American Economic Review 2018, 108(6): 1488–1542

https://doi.org/10.1257/aer.20160696

The Race between Man and Machine: Implications of

Technology for Growth, Factor Shares, and Employment†
By Daron Acemoglu and Pascual Restrepo*

We examine the concerns that new technologies will render labor

redundant in a framework in which tasks previously performed
by labor can be automated and new versions of existing tasks, in
which labor has a comparative advantage, can be created. In a
static v­ ersion where capital is fixed and technology is exogenous,
­automation reduces employment and the labor share, and may even
reduce wages, while the creation of new tasks has the opposite effects.
Our full model endogenizes capital accumulation and the direction
of research toward automation and the creation of new tasks. If the
long-run rental rate of capital relative to the wage is sufficiently low,
the long-run equilibrium involves automation of all tasks. Otherwise,
there exists a stable balanced growth path in which the two types
of innovations go hand-in-hand. Stability is a c­ onsequence of the
fact that automation reduces the cost of p­ roducing using labor, and
thus discourages further automation and encourages the c­ reation
of new tasks. In an extension with heterogeneous skills, we show
that i­nequality increases during transitions driven both by faster
­automation and the introduction of new tasks, and characterize
the conditions under which inequality stabilizes in the long run.
(JEL D63, E22, E23, E24, J24, O33, O41)

The accelerated automation of tasks performed by labor raises concerns that new
technologies will make labor redundant (e.g., Brynjolfsson and McAfee 2014; Akst
2013; Autor 2015). The recent declines in the labor share in national income and
the employment to population ratio in the United States (e.g., Karabarbounis and
Neiman 2014; Oberfield and Raval 2014) are often interpreted as e­ vidence for the
claims that as digital technologies, robotics, and artificial intelligence p­ enetrate the
economy, workers will find it increasingly difficult to compete against machines,
and their compensation will experience a relative or even absolute decline.

* Acemoglu: Department of Economics, MIT, 77 Massachusetts Avenue, Cambridge, MA 02142

(email: daron@mit.edu); Restrepo: Department of Economics, Boston University, 270 Bay State Road, Boston,
MA 02215 (email: pascual@bu.edu). This paper was accepted to the AER under the guidance of Mark Aguiar,
Coeditor. We thank Philippe Aghion, David Autor, Erik Brynjolfsson, Chad Jones, John Van Reenen, three anon-
ymous referees, and participants at various conferences and seminars for useful comments and suggestions. We
are grateful to Giovanna Marcolongo and Mikel Petri for research assistance. Jeffrey Lin generously shared all
of his data with us. We also gratefully acknowledge financial s­ upport from the Bradley Foundation, the Sloan
Foundation, and the Toulouse Network on Information Technology. Restrepo thanks the Cowles Foundation and
the Yale Economics Department for their hospitality.
†
Go to https://doi.org/10.1257/aer.20160696 to visit the article page for additional materials and author
disclosure statement(s).

1488
VOL. 108 NO. 6 ACEMOGLU AND RESTREPO: THE RACE BETWEEN MAN AND MACHINE 1489

Yet, we lack a comprehensive framework incorporating such effects, as well as

potential countervailing forces.
The need for such a framework stems not only from the importance of
­understanding how and when automation will transform the labor market, but also
from the fact that similar claims have been made, but have not always come true,
about previous waves of new technologies. Keynes famously foresaw the steady
increase in per capita income during the twentieth century from the introduction
of new technologies, but incorrectly predicted that this would create widespread
technological unemployment as machines replaced human labor (Keynes 1930).
In 1965, economic historian Robert Heilbroner confidently stated that “as machines
continue to invade society, duplicating greater and greater numbers of social tasks,
it is human labor itself—at least, as we now think of ‘labor’— that is gradually
­rendered redundant” (quoted in Akst 2014, p. 2). Wassily Leontief was equally pes-
simistic about the ­implications of new machines. By drawing an analogy with the
technologies of the early twentieth century that made horses redundant, in an inter-
view1 he speculated that “Labor will become less and less important... More and
more workers will be replaced by machines. I do not see that new industries can
employ everybody who wants a job.”
This paper is a first step in developing a conceptual framework to study how
machines replace human labor and why this might (or might not) lead to lower
employment and stagnant wages. Our main conceptual innovation is to propose a
framework in which tasks previously performed by labor are automated, while at
the same time other new technologies complement labor : specifically, in our model
this takes the form of the introduction of new tasks in which labor has a comparative
advantage. Herein lies our answer to Leontief’s analogy: the difference between
human labor and horses is that humans have a comparative advantage in new and
more complex tasks. Horses did not. If this comparative advantage is significant and
the creation of new tasks continues, employment and the labor share can remain
stable in the long run even in the face of rapid automation.
The importance of new tasks is well illustrated by the technological and
­organizational changes during the Second Industrial Revolution, which not only
involved the replacement of the stagecoach by the railroad, sailboats by steamboats,
and of manual dock workers by cranes, but also the creation of new labor-intensive
tasks. These tasks generated jobs for engineers, machinists, repairmen, conductors,
back-office workers, and managers involved with the introduction and operation of
new technologies (e.g., Landes 1969; Chandler 1977; and Mokyr 1990).
Today, as industrial robots, digital technologies, computer-controlled machines,
and artificial intelligence replace labor, we are again witnessing the emergence of
new tasks ranging from engineering and programming functions to those performed
by audio-visual specialists, executive assistants, data administrators and analysts,
meeting planners, and social workers. Indeed, during the last 35 years, new tasks
and new job titles have accounted for a large fraction of US employment growth.
To document this fact, we use data from Lin (2011) to measure the share of new job
titles, jobs in which workers perform tasks that are different from tasks in previously

1
Charlotte Curtis, “Machines vs. Workers,” The New York Times, February 8, 1983.
1490 THE AMERICAN ECONOMIC REVIEW JUNE 2018

10%

Employment growth 1980–2015

5%

0%

−5%

0 0.2 0.4 0.6 0.8

Share of new job titles in 1980

Figure 1. Employment Growth by Occupation between 1980 and 2015 (Annualized)

and the Share of New Job Titles in 1980

existing jobs , within each occupational category. In 2000, about 70 percent of com-
puter software developers (an occupational category employing one million people
at the time) held new job titles. Similarly, in 1990 “radiology technician” and in 1980
“management analyst” were new job titles. Figure 1 shows that occupations with
10 percentage points more new job titles (which is approximately the ­sample ­average
in 1980) experienced 0.41 percent faster employment growth between 1980 and
2015. This estimate implies that about 60 percent of the 50 million or so jobs added
during this 35-year period are associated with the additional employment growth in
occupations with new job titles (relative to occupations with no new job titles).2
We start with a static model in which capital is fixed and technology is
­exogenous. There are two types of technological changes: automation allows firms
to ­substitute capital for tasks previously performed by labor, while the creation
of new tasks enables the replacement of old tasks by new variants in which labor
has a higher ­productivity. Our static model provides a rich but tractable frame-
work that c­ larifies how automation and the creation of new tasks shape the pro-
duction p­ ossibilities of the economy and determine factor prices, factor shares in
national income, and employment. Automation always reduces the labor share and
­employment, and may even reduce wages.3 Conversely, the creation of new tasks

2
The relationship shown in Figure 1 controls for the demographic composition of employment in the ­occupation
in 1980. In online Appendix B, we show that the same relationship holds between the share of new job titles in
1990 (in 2000) and employment growth from 1990 to 2015 (from 2000 to 2015), and that these patterns are p­ resent
­without any controls and when we control for average education in the occupation and the structural changes
in the US ­economy as well. The data for 1980, 1990 and 2000 are from the US Census. The data for 2015 are
from the American Community Survey. Additional information on the data and our sample is provided in online
Appendix B.
3
The effects of automation in our model contrast with the implications of factor-augmenting technologies.
As we discuss in greater detail later and in particular in footnote 19, the effects of factor-augmenting technologies
VOL. 108 NO. 6 ACEMOGLU AND RESTREPO: THE RACE BETWEEN MAN AND MACHINE 1491

increases wages, ­employment, and the labor share. These comparative statics follow
because factor prices are determined by the range of tasks performed by capital and
labor, and shifts in technology alter the range of tasks performed by each factor
(see also Acemoglu and Autor 2011).
We then embed this framework in a dynamic economy in which capital
­accumulation is endogenous, and we characterize restrictions under which the
model delivers balanced growth with automation and creation of new tasks , which
we take to be a good approximation to economic growth in the United States and
the United Kingdom over the last two centuries. The key restrictions are that there
is exponential productivity growth from the creation of new tasks and that the two
types of technological changes (automation and the creation of new tasks) advance
at equal rates. A critical difference from our static model is that capital ­accumulation
responds to permanent shifts in technology in order to keep the interest rate
and hence the rental rate of capital constant. As a result, the dynamic effects of
­technology on factor prices depend on the response of capital a­ ccumulation as well.
The response of capital ensures that the productivity gains from both a­ utomation and
the ­introduction of new tasks fully accrue to labor (the relatively inelastic f­actor).
Although the real wage in the long run increases because of this productivity effect,
automation still reduces the labor share and employment.
Our full model endogenizes the rates of improvement of these two types of
­technologies by marrying our task-based framework with a directed technological
change setup. This full version of the model remains tractable and allows a complete
characterization of balanced growth paths. If the long-run rental rate of capital is very
low relative to the wage, there will not be sufficient incentives to create new tasks,
and the long-run equilibrium involves full automation, akin to Leontief’s “ horse
equilibrium.” Otherwise, the long-run equilibrium involves ­balanced growth based
on equal advancement of the two types of technologies. Under natural ­assumptions,
this (interior) balanced growth path is stable, so that when ­automation runs ahead
of the creation of new tasks, market forces induce a slowdown in ­subsequent
­automation and more rapid countervailing advances in the creation of new tasks.
This stability result highlights a crucial new force: a wave of automation pushes
down the ­effective cost of producing with labor, discouraging further efforts to
­automate additional tasks and encouraging the creation of new tasks.
The stability of the balanced growth path implies that periods in which ­automation
runs ahead of the creation of new tasks tend to trigger self-correcting forces, and
as a result, the labor share and employment stabilize and could return to their
­initial ­levels. Whether this is the case depends on the reason why automation paced
ahead in the first place. If this is caused by the random arrival of a series of automa-
tion technologies, the long-run equilibrium takes us back to the same initial levels
of employment and labor share. If, on the other hand, automation surges because of
a change in the innovation possibilities frontier (making automation easier relative
to the creation of new tasks), the economy will tend toward a new balanced growth

on the labor share depend on the elasticity of substitution between capital and labor. In addition, ­capital-augmenting
­technological improvements always increase the wage, while labor-augmenting ones also increase the wage
­provided that the elasticity of substitution between capital and labor is greater than the capital share in national
income. This contrast underscores that it would be misleading to think of automation in terms of factor-augmenting
technologies. See Acemoglu and Restrepo (2018).
1492 THE AMERICAN ECONOMIC REVIEW JUNE 2018

path with lower levels of employment and labor share. In neither case does rapid
automation necessarily bring about the demise of labor.4
We also consider three extensions of our model. First, we introduce heterogeneity
in skills, and assume that skilled labor has a comparative advantage in new tasks,
which we view as a natural assumption.5 Because of this pattern of comparative
advantage, automation directly takes jobs away from unskilled labor and increases
inequality, while new tasks directly benefit skilled workers and at first increase
inequality as well. Over the long run, the standardization of new tasks help low-skill
workers. We characterizes the conditions under which standardization is sufficient
to restore stable inequality in the long run. This extension formalizes the idea that
both automation and the creation of new tasks increase inequality in the short run
but standardization limits the increase in inequality in the long run.
Our second extension modifies our baseline patent structure and reintroduces the
creative destruction of the profits of previous innovators, which is absent in our main
model, though it is often assumed in the endogenous growth literature. The results in
this case are similar, but the conditions for uniqueness and stability of the balanced
growth path are more demanding.
Finally, we study the efficiency properties of the process of automation and
creation of new technologies, and point to a new source of inefficiency leading to
excessive automation: when the wage rate is above the opportunity cost of labor
(due to labor market frictions), firms will choose automation to save on labor costs,
while the social planner, taking into account the lower opportunity cost of labor,
would have chosen less automation.
Our paper can be viewed as a combination of task-based models of the labor
­market with directed technological change models.6 Task-based models have been
­developed both in the economic growth and labor literatures, dating back at least to
Roy’s (1951) seminal work. The first important recent contribution, Zeira (1998),
proposed a model of economic growth based on capital-labor s­ubstitution.
Zeira’s model is a special case of our framework. Acemoglu and Zilibotti (2001)
­developed a simple task-based model with endogenous technology and applied it
to the study of productivity differences across countries, illustrating the ­potential
mismatch between new technologies and the skills of developing economies
­
(see also Zeira 2006; Acemoglu 2010). Autor, Levy, and Murnane (2003) suggested
that the increase in inequality in the US labor market reflects the automation and
­computerization of routine tasks.7 Our static model is most similar to Acemoglu
and Autor (2011). Our full framework extends this model not only because of the
dynamic e­ quilibrium ­incorporating capital accumulation and directed technological
change, but also because tasks are combined with a general elasticity of substitution,

4
Yet, it is also possible that some changes in parameters shift us away from the region of stability to the full
automation equilibrium.
5
This assumption builds on Schultz (1975). See also Greenwood and Yorukoglu (1997); Caselli (1999); Galor
and Moav (2000); Acemoglu, Gancia, and Zilibotti (2012); and Beaudry, Green, and Sand (2016).
6
On directed technological change and related models, see Acemoglu (1998, 2002, 2003, 2007); Kiley (1999);
Caselli and Coleman (2006); Thoenig and Verdier (2003); and Gancia, Müller, and Zilibotti (2013).
7
Acemoglu and Autor (2011); Autor and Dorn (2013); Jaimovich and Siu (2014); Foote and Ryan (2015);
Burstein, Morales, and Vogel (2014); and Burstein and Vogel (2017) provide various pieces of empirical evidence
and quantitative evaluations on the importance of the endogenous allocation of tasks to factors in recent labor
market dynamics.
VOL. 108 NO. 6 ACEMOGLU AND RESTREPO: THE RACE BETWEEN MAN AND MACHINE 1493

and because the equilibrium a­ llocation of tasks depends both on factor prices and
the state of technology.8
Three papers from the economic growth literature that are related to our work
are Acemoglu (2003), Jones (2005), and Hémous and Olsen (2016). The first
two papers develop growth models in which the aggregate production f­unction is
endogenous and, in the long run, adapts to make balanced growth ­possible. In Jones
(2005), this occurs because of endogenous choices about different c­ ombinations
of ­activities/­technologies. In Acemoglu (2003), this long-run behavior is a
­consequence of directed technological change in a model of factor-augmenting
technologies. Our task-based framework here is a significant departure from this
model, e­ specially since it enables us to address questions related to automation, its
impact on factor prices and its endogenous evolution. In addition, our ­framework
provides a more robust economic force ensuring the stability of the balanced growth
path: while in models with f­actor-augmenting technologies stability requires an
elasticity of s­ ubstitution between capital and labor that is less than 1 (so that the
more ­abundant factor commands a lower share of national income), we do not need
such a condition in this framework.9 Hémous and Olsen (2016) propose a model
of a­ utomation and horizontal innovation with endogenous technology, and use it
to study the ­consequences of different types of technologies on inequality. High
wages (in their model for low-skill workers) encourage automation. But unlike
in our model, the unbalanced dynamics that this generates are not countered by
other types of innovations in the long run. Also worth noting is Kotlikoff and Sachs
(2012), who develop an overlapping generation model in which automation may
have ­long-lasting effects. In their model, automation reduces the earnings of current
workers, and via this channel, depresses savings and capital accumulation.
The rest of the paper is organized as follows. Section I presents our ­task-based­
framework in the context of a static economy. Section II introduces capital
­
­accumulation and clarifies the conditions for balanced growth in this economy.
Section III presents our full model with endogenous technology and establishes,
under some plausible conditions, the existence, uniqueness, and stability of a
­balanced growth path with two types of technologies advancing in tandem. Section IV
­considers the three extensions mentioned above. Section V concludes. Appendix A
contains the proofs of our main results, while online Appendix B contains the
remaining proofs, additional results, and the details of the empirical analysis pre-
sented above.

I. Static Model

We start with a static version of our model with exogenous technology, which
allows us to introduce our main setup in the simplest fashion and characterize the

8
Acemoglu and Autor’s model, like ours, is one in which a discrete number of labor types are allocated to a con-
tinuum of tasks. Costinot and Vogel (2010) develop a complementary model in which there is a continuum of skills
and a continuum of tasks. See also Hawkins, Michaels, and Oh (2015), which shows how a task-based model is
more successful than standard models in matching the comovement of investment and ­employment at the firm level.
9
The role of technologies replacing tasks in this result can also be seen by noting that with factor-augmenting
technologies, the direction of innovation may be dominated by a strong market size effect (e.g., Acemoglu 2002).
Instead, in our model, it is the difference between factor prices that regulates the future path of technological
change, generating a powerful force toward stability.
1494 THE AMERICAN ECONOMIC REVIEW JUNE 2018

impact of different types of technological change on factor prices, employment, and

the labor share.

A. Environment

The economy produces a unique final good Y ​ ​by combining a unit measure of
tasks, ​y(i)​ , with an elasticity of substitution ​σ ∈ (0, ∞)​:

​  σ ​
_

B​​(∫
​ N−1​​​ y ​(i)​​  ​ σ ​​di)​​​  ​,​
~ N σ−1
_ σ−1
(1) ​
Y = ​

~
where ​​B​ > 0​. All tasks and the final good are produced competitively. The fact
that the limits of integration run between ​N − 1​and N
​ ​imposes that the measure of
tasks used in production always remains at 1. A new (more complex) task replaces
or upgrades the lowest-index task. Thus, an increase in N ​ ​represents the upgrading
of the quality (productivity) of the unit measure of tasks.10
Each task is produced by combining labor or capital with a task-specific
­intermediate ​q(i)​ , which embodies the technology used either for automation or
for production with labor. To simplify the exposition, we start by assuming that
these intermediates are supplied competitively, and that they can be produced using​
ψ​units of the final good. Hence, they are also priced at ​ψ​. In Section III we relax
this assumption and allow intermediate producers to make profits so as to generate
­endogenous incentives for innovation.
All tasks can be produced with labor. We model the technological constraints
on automation by assuming that there exists I​ ∈ [N − 1, N]​such that tasks i​ ≤ I​
are technologically automated in the sense that it is feasible to produce them with
capital. Although tasks ​i ≤ I​are technologically automated, whether they will be
produced with capital or not depends on relative factor prices as we describe below.
Conversely, tasks ​i > I​are not technologically automated, and must be produced
with labor.
The production function for tasks i​ > I​takes the form

ζ
_
_ ​  ​
y(i) = ​B ​(ζ)​​[​η​​  ​  ζ ​​q ​(i)​​  ​ ζ ​​+ ​(1 − η)​​  ​  ζ ​​(​​ γ(i) l(i))​​​  ​ ζ ​​]​​​  ​,​
_ ζ−1
_ _ ζ−1 ζ−1
_
(2) ​
1 1

where ​γ(i)​denotes the productivity of labor in task ​i​  , ​ζ ∈ (0, ∞)​is the elasticity
of substitution between intermediates and labor, η​ ∈ (0, 1)​is the share
­parameter_ of this constant elasticity of substitution (CES) production function,
and
_ ​  ​
B (
​ ζ)​is a ­constant included to simplify_ the algebra. In particular, we set
​​  B ​(ζ ) = ​ψ​​  η​​(1 − η)​​  η−1​​η​​  −η​​when ζ​ = 1​, and ​​  B ​(ζ ) = 1​ otherwise.

10
This formulation imposes that once a new task is created at ​N​it will be immediately utilized and replace the
​ − 1​. This is ensured by Assumption 3, and avoids the need for additional notation
lowest available task located at N
at this point. We view newly-created tasks as higher productivity versions of existing tasks.
VOL. 108 NO. 6 ACEMOGLU AND RESTREPO: THE RACE BETWEEN MAN AND MACHINE 1495

Tasks ​i ≤ I​can be produced using labor or capital, and their production ­function
is identical to (2) except for the presence of capital and labor as perfectly s­ ubstitutable
factors of production:11
ζ
_
_ ​  ​
y(i) = ​  B ​(ζ)​​[​η​​  ​  ζ ​​q ​(i)​​  ​ ζ ​​+ ​(1 − η)​​  ​  ζ ​​​​(k(i) + γ(i) l(i))​​​  ​ ζ ​​]​​​  ​.​
_ ζ−1
_ _ ζ−1 ζ−1
_
(3) ​
1 1

Throughout, we impose the following assumption.

ASSUMPTION 1: γ
​ (i)​is strictly increasing.

Assumption 1 implies that labor has strict comparative advantage in tasks with a
higher index and will guarantee that, in equilibrium, tasks with lower indices will be
automated, while those with higher indices will be produced with labor.
We model the demand side of the economy using a representative household with
preferences given by
−ν  (L) 1− θ
​(C​e​​  ​)​​  ​−1
____________
(4) ​
u(C, L) = ​     ​ ,​
1−θ
where ​C​is consumption, ​L​denotes the labor supply of the representative ­household,
and ​ ν(L)​designates the utility cost of labor supply, which we assume to be
­continuously differentiable, increasing, and convex, and to satisfy ν​ ″(L) + (θ − 1)​
× (ν ′(L))​​  2​/θ > 0​ (which ensures that u​ (C, L)​is concave). The functional form
in (4) ensures balanced growth (see King, Plosser, and Rebelo 1988; Boppart and
Krusell 2016). When we turn to the dynamic analysis in the next section, θ​ ​will be
the inverse of the intertemporal elasticity of substitution.
Finally, in the static model, the capital stock, ​K​ , is taken as given (it will be
endogenized via household saving decisions in Section II).

B. Equilibrium in the Static Model

Given the set of technologies I​ ​and N

​ ​ , and the capital stock K
​ ​ , we now c­ haracterize
the equilibrium value of output, factor prices, employment, and the threshold task ​I *​.
In the text, we simplify the exposition by imposing the following assumption.

ASSUMPTION 2: One of the following two conditions holds:

(i) ​η → 0​  , or

(ii) ​ζ = 1​.

These two special cases ensure that the demand for labor and capital is
­homothetic. More generally, our qualitative results are identical as long as the

11
A simplifying feature of the technology described in equation (3) is that capital has the same ­productivity in all
tasks. This assumption could be relaxed with no change to our results in the static model, but without other changes,
it would not allow balanced growth in the next section. Another simplifying assumption is that ­non-automated tasks
can be produced with just labor. Having these tasks combine labor and capital would have no impact on our main
results as we show in online Appendix B.
1496 THE AMERICAN ECONOMIC REVIEW JUNE 2018

degree of ­non-homotheticity is not too extreme, though in this case we no longer

have ­closed-form expressions and this motivates our choice of presenting these
more ­general results in Appendix A.12
We proceed by characterizing the unit cost of producing each task as a function
of factor prices and the automation possibilities represented by ​I.​ Because tasks are
produced competitively, their price, ​p(i)​, will be equal to the minimum unit cost of
production:

⎧
⎪min​​{R, _ γ(i) }
1−η
​ if i ≤ I ​ ​  W ​ ​​​ 
(5) ​ ⎨
p(i) = ​ ​    ​  ​  ​  ​​​ ​,
⎪​​ _
⎩( γ(i) )
1−η
​  W ​ ​​​  ​ if i > I ​

where ​W​denotes the wage rate and ​R​denotes the rental rate of capital.
In equation (5), the unit cost of production for tasks ​i > I​is given by the
effective cost of labor, ​W/γ(i)​ (which takes into account that the productivity of
labor in task i​​is ​γ(i)​). The unit cost of production for tasks i​ ≤ I​  is given by

{ γ(i) }
​min​ R, _
​  W ​ ​​, which reflects the fact that capital and labor are perfect substitutes in
the production of automated tasks. In these tasks, firms will choose whichever factor
has a lower effective cost:  ​R​or W ​ /γ(i)​.
Because labor has a strict comparative advantage in tasks with a higher index,
~
there is a (unique) threshold ​​ I​​ such that

()
W ​ = γ​ ~​​ I ​.​
​ _
(6) ​
R
This threshold represents the task for which the costs of producing with capital
~
and labor are equal. For all tasks ​i ≤ ​ ​​ I, we have ​R ≤ W/γ(i)​, and without any
~
other constraints, these tasks will be produced with capital. However, if I​ ​​  > I​, firms
~
­cannot produce all tasks until ​​ I​​with capital because of the constraint imposed by the
available automation technology. This implies that there exists a unique equilibrium
threshold task

I * = min ​{I, ​ ​I}​​,
~
​

12
The source of non-homotheticity in the general model is the substitution between factors (­capital or
labor) and intermediates (the q​(i)​s). A strong substitution creates implausible features. For example,
­automation, which increases the price of capital, may end up raising the demand for labor more than the
demand for capital, as capital gets substituted by the intermediate inputs. Assumption 2​ ′​in Appendix A imposes
max{1, σ}
γ(N − 1)
( γ(N) )
that ​​​ _
​  ​______________
​ ​​​ 
​     1
|1−ζ |
​ > |σ − ζ |​which ensures that the degree of non-homotheticity is not
γ (N)
( γ (N − 1) )
​​ ​  _ ​ ​​​  ​− 1

too extreme and automation always reduces the relative demand for labor.
VOL. 108 NO. 6 ACEMOGLU AND RESTREPO: THE RACE BETWEEN MAN AND MACHINE 1497

Panel A
~
N−1 I* = I I N

Tasks performed by capital Labor-intensive tasks

Panel B
~
N−1 I* = I I N N

Capital Labor New tasks

Replaced tasks

Panel C
~
N−1 I* I* = I I N

Tasks performed by capital Labor-intensive tasks

Automated tasks

Figure 2. The Task Space and a Representation of the Effect of Introducing New
Tasks (Panel B) and Automating Existing Tasks (Panel C )

such that all tasks ​i ≤ I *​will be produced with capital, while all tasks ​i > I *​ will
be produced with labor.13
Figure 2 depicts the resulting allocation of tasks to factors and also shows how,
as already noted, the creation of new tasks replaces existing tasks from the bottom
of the distribution.
As noted in footnote 10, we have simplified the exposition by imposing that new
tasks created at N ​ ​immediately replace tasks located at N ​ − 1​, and it is therefore
profitable to produce new tasks with labor (and hence we have not distinguished ​N​  ,​​
˜
N​​  ∗​​, and N​
​​ ​). In the static model, this will be the case when the capital stock is not too
large, which is imposed in the next assumption.
_ _
​ < ​  K ​​, where ​​  K ​​is such that ​R = _
ASSUMPTION 3: We have K ​  W ​​.
γ(N)

This assumption ensures that ​ R > ​_ W ​ ,​ and consequently, new tasks will
γ (N)
increase aggregate output and will be adopted immediately. Outside of this region,
new tasks would not be utilized, which we view as the less interesting case. This
assumption is relaxed in the next two sections where the capital stock is endogenous.
We next derive the demand for factors in terms of the (endogenous) threshold​
I *​and the technology parameter N
​ ​. We choose the final good as the numéraire.
Equation (1) gives the demand for task i​​ as

13
Without loss of generality, we impose that firms use capital when they are indifferent between using capital
or labor, which explains our convention of writing that all tasks ​i ≤ I *​ (rather than ​i < I *​) are produced using
capital.
1498 THE AMERICAN ECONOMIC REVIEW JUNE 2018

~σ−1
(7) ​
y(i) = ​​ B​​​  ​  Yp​(i)​​  −σ​.​
~____ ​  σ−1 ​
Let us define ​​ ˆ
σ​ = σ(1 − η) + ζη​and ​ B = ​​ B​​​  σ​​ˆ−1 ​​. Under Assumption 2,
­equations (2) and (3) yield the demand for capital and labor in each task as

{0
​ˆ−1 ˆ​ if i ≤ I *
​B​​  σ​ ​(1 − η) ​YR​​  −​σ​
k(i ) = ​ ​   
​ ​  ​  ​,​​
if i > I *
and
⎧0 if i ≤ I *
l(i) = ⎨
⎪

​ ​    
​​ˆ−1 ​ ​ 
ˆ
− σ​
​.​​​
γ(i) ( γ(i) )
⎪​B​​  σ​ ​(1 − η)Y ​ ___
1  ​ ​​ ___
​  W  ​ ​​​  ​ if i > I *
⎩

We can now define a static equilibrium as follows. Given a range of tasks​

[N − 1, N ]​, automation technology I​ ∈ (N − 1, N ]​, and a capital stock K ​ ​ , a static
~
equilibrium is summarized by a set of factor prices, W ​ ​and R​ ​  , threshold tasks, I​​​  ​ and​
I *​, employment level, L
​ ​ , and aggregate output, ​Y​ , such that

• ​​ I​​is determined by equation (6) and I​ * = min​{I, ​ I​}​​;

~ ~
• The capital and labor markets clear, so that
​ˆ
(8) ​​B​​  σ​−1 ​(1 − η) Y(I * − N + 1)​R​​  −​ˆσ​​ = K, ​

(9) ​​B​​  ​ˆσ​−1​(1 − η) Y ∫

−​ˆ
σ​

I γ (i) ( γ (i) )
​ *​  ​​ ​ _
1 ​ ​​ _
​  W ​ ​​​  ​di = L; ​
N

• Factor prices satisfy the ideal price index condition,

(I * − N + 1) ​R​​  1−​ˆσ​​+ ​∫ *​ ​​ ​​ _
ˆ
1−​σ​

I ( γ(i) )
(10) ​ ​  W ​ ​​​  ​di = ​B​​  1−​ˆσ​​; ​
N

• Labor supply satisfies ​ν ′(L) = W/C​. Since in equilibrium ​C = RK + WL​  ,

this condition can be rearranged to yield the following increasing labor supply
function:14

L = ​L​​  s​​(_
(11) ​ ​  W )
​ ​.​
RK
PROPOSITION 1 (Equilibrium in the Static Model): Suppose that Assumptions 1,
2, and 3 hold. Then a static equilibrium exists and is unique. In this static equilib-
rium, aggregate output is given by
ˆ
​  ​
​
σ
​ ____

[ ​  di)​​​  ​​L​​  ​  σ​​ˆ ​​]​​​ 

__
​1  ​ ​ˆ
​−1
σ
Y = ​ _
(12) ​
ˆ
​  ​
​  σ
​−1
B  ​ ​​ ​(I * − N + 1)​​  __​  ​σ​ˆ1 ​​​K​​  ____
1−η
​ˆ
σ ∫I
​ ​+ ​​(​ *​  ​​ γ ​(i)​​ 
​ˆ
​−1
σ
N ˆ ____ ˆ
​σ
​
​
​−1
σ
​.​

14
This representation clarifies that the equilibrium implications of our setup are identical to one in which
an upward-sloping quasi-labor supply determines the relationship between employment and wages (and does not
necessarily equate marginal cost of labor supply to the wage). This follows readily by taking (11) to represent this
quasi-labor supply relationship.
VOL. 108 NO. 6 ACEMOGLU AND RESTREPO: THE RACE BETWEEN MAN AND MACHINE 1499

Panel A Panel B
min{I, I} min{I′, I}
~ ~
min{I, I} min{I′, I}
~ ~
ω ω
γ(I) = ωK γ(I) = ωK
~ ~
↓
ω′

ω = ω(I *, N, K)

ω = ω(I *, N, K)

~
I * = I → I * = I′ i I* = I i

Figure 3. Static Equilibrium

~
Notes: Panel A depicts the case in which ​​​I​​ *​  = I < ​  ​I​​​​so that the allocation of factors is constrained by technology.
~
Panel B depicts the case in which ​​​I​​ *​  = ​ ​I​​ < I​​so that the allocation of factors is not constrained by technology and
is cost-minimizing. The blue curves show the shifts following an increase in ​​I​​ to ​​​I′​, which reduce ω ​ ​in the panel A,
but have no effect in panel B.

PROOF:
See Appendix A.

Equation (12) shows that aggregate output is a CES aggregate of capital and
labor, with the elasticity between capital and labor being ​​ˆ σ​​. The share ­parameters are
endogenous and depend on the state of the two types of technologies and the equilib-
rium choices of firms. An increase in I​ *​, which corresponds to greater e­ quilibrium
automation , increases the share of capital and reduces the share of labor in this
aggregate production function, while the creation of new tasks does the opposite.
Figure 3 illustrates the unique equilibrium described in Proposition 1.
The ­equilibrium is given by the intersection of two curves in the (​ ω, I)​ space, where​
ω = ​_ W ​​is the wage level normalized by capital income; this ratio is a ­monotone
RK
transformation of the labor share and will play a central role in the rest of our
­analysis.15 The upward-sloping curve represents the cost-minimizing allocation of
capital and labor to tasks represented by equation (6), with the constraint that the
equilibrium level of automation can never exceed I​ ​. The downward-sloping curve,​
ω( I *, N, K)​ , corresponds to the relative demand for labor, which can be obtained
directly from (8), (9), and (11) as
∫
​ *​  ​​ ​γ (i)​​  ​ˆσ​−1​  di
N

(13) ​ln ω + __ (​  ​ˆσ​ ​ − 1)​  ln K + ​  ​ˆσ​ ​ ln​(​    I* − N + 1 )

​  1  ​ ln ​L​​  s​(ω) = ​ __ 1 __ 1 __________
I
  
​ˆσ​
 ​ ​.​

As we show in Appendix A, the relative demand curve always starts above the cost
minimization condition and ends up below it, so that the two curves necessarily
intersect, defining a unique equilibrium as shown in Figure 3.
The figure also distinguishes between the two cases highlighted above. In panel
~
A, we have ​I * = I < ​ ​​
I and the allocation of factors is constrained by t­echnology,

15
The increasing labor supply relationship, (11), ensures that the labor share ​s​​L​ = ​ ______
WL
RK + WL
 ​​is increasing
in ω
​ ​.
1500 THE AMERICAN ECONOMIC REVIEW JUNE 2018

~
while panel B plots the case where I​ * = ​ ​I < I​and firms choose the cost-minimiz-
ing allocation given factor prices.
A special case of Proposition 1 is also worth highlighting, because it leads to a
Cobb-Douglas production function with an exponent depending on the degree of
automation, which is particularly tractable in certain applications.

COROLLARY 1: Suppose that ​σ = ζ = 1​and ​γ (i) = 1​for all ​i​. Then aggregate
output is

Y = ​ _
​ B  ​ ​K​​  1−N+I* ​​L​​  N−I*.​
​
1−η
The next two propositions give a complete characterization of comparative statics.16

PROPOSITION 2 (Comparative Statics): Suppose that Assumptions 1, 2, and 3

hold. Let ​​εL​ ​ > 0​denote the elasticity of the labor supply schedule ​​L​​  s​(ω)​; let​​
d ln γ (I)
ε​γ​​ = _
​ ​ > 0​denote the semi-elasticity of the comparative advantage
dI
­schedule; and let
​ˆ−1
​γ (I *)​​  σ​ ​ ​γ (N)​​  ​ˆσ​−1​
​ ​Λ​I​ = ​ __________
   ​ + ​ _______ 1  ​  and ​
Λ ​ ​ = ​  _________
    ​ + ​ ________
1  ​.​
​∫I*​  ​​ γ ​(i)​​ 
N ˆ
σ​
​ − 1 ​di I *−N+1 N
∫
​ N​  ​​ γ ​(i)​​  ​ˆ
σ​−1
​
d i I *−N+1
I*
~
• If ​I * = I < ​ I​​  , so that the allocation of tasks to factors is constrained by
technology, then:

(i) the impact of technological change on relative factor prices is given by

_ d ln (W/R) d ln ω ​ =
​  ​ = ​ _ −  ​ _____ 1  ​  ​Λ​ ​​ < 0,
​ˆσ​ + ​εL​ ​​ I
​
dI dI
_ d ln (W/R) d ln ω ​ =
​  ​ = ​ _ _____
​  1  ​   ​ΛN​ ​​ > 0;​
​ˆσ​ + ​εL​ ​​
 
dN dN
(ii) and the impact of capital on relative factor prices is given by

d ln (W/R) 1 + ​εL​ ​​
d ln ω ​+ 1 = ​ _____
​ _​ = ​ _  ​  > 0.​
​ˆσ​ + ​εL​ ​​
​
d ln K d ln K
~
• If ​I *  =  ​ I​  <  I​, so that the allocation of tasks to factors is cost-minimizing, then:

(i) the impact of technological change on relative factor prices is given by

d ln (W/R) d ln (W/R)
_
​ 
​ ​  d ln ω ​ = 0,
​= _ ​ _​ = _ ​  d ln ω ​ = ​ _
​ ​free​​ + ​εL​  ​ ​​ ​ΛN​ ​​ > 0,​
σ
1
dI dI dN dN
where
​ ˆ+_
​σ​free​​ = ​σ​ ​  ​ε1​ ​​ ​  ​ΛI​ ​​ > ​ˆ
γ
σ​;​

~
16
In this proposition, we do not explicitly treat the case in which ​I * = I = ​ I​​in order to save on space and
notation, since in this case left and right derivatives with respect to ​I​are different.
VOL. 108 NO. 6 ACEMOGLU AND RESTREPO: THE RACE BETWEEN MAN AND MACHINE 1501

(ii) and the impact of capital on relative factor prices is given by

_ d ln (W/R) d ln ω ​+ 1 = _ 1 + ​εL​ ​​
​ 
​ ​ = ​ _ ​ ​free​​ + ​εL​  ​ ​​ > 0.​
​  σ
d ln K d ln K
• In all cases, the labor share and employment move in the same direction as​

ω:​ _
​​dL ​ > 0​ and, when ​I * = I​, _
​​dL ​ < 0​.
dN dI

PROOF:
See online Appendix B.

The main implication of Proposition 2 is that the two types of t­echnological

change (automation and the creation of new tasks) have polar implications.
An increase in N ​ ​  ( the creation of new tasks) raises ​W/R​ , the labor share, and
employment. An increase in I​ ​  (an improvement in automation technology) reduces​
~
W/R​ , the labor share, and employment (unless I​ * = ​ I​ < I​and firms are not con-
strained by technology in their automation choice).17
~
The reason why automation reduces employment (when I​ * = I < ​ I​​  ) is that it
raises aggregate output per worker more than it raises wages (as we will see next,
automation may even reduce wages). Thus, the negative income effect on the labor
supply resulting from greater aggregate output dominates any substitution effect that
might follow from the higher wages. On the other hand, the creation of new tasks
always increases employment: new tasks raise wages more than aggregate output,
increasing the labor supply. Although these exact results rely on the balanced growth
preferences in equation (4), similar forces operate in general and create a tendency
for automation to reduce employment and for new tasks to increase it.
Figure 3 illustrates the comparative statics: automation moves us along the
­relative labor demand curve in the technology-constrained case shown in panel A
(and has no impact in panel B), while the creation of new tasks shifts out the relative
labor demand curve in both cases.
A final implication of Proposition 2 is that the “technology-constrained” e­ lasticity
~
of substitution between capital and labor, σ​ ˆ
​​ ​, which applies when I​ * = I < ​ I​​, ­differs
from the “technology-free” elasticity, ​σ ​ f​ ree​​​ , which applies when the decision of
~
which tasks to automate is not constrained by technology (i.e., when I​ * = ​ I​ < I​  ).
This is because in the former case, as relative factor prices change, the set of tasks
performed by each factor remains fixed. In the latter case, as relative factor prices
change, firms reassign tasks to factors. This additional margin of adjustment implies
​ f​ ree​​ > ​
that ​σ ˆ
σ​​.

PROPOSITION 3 (Impact of Technology on Productivity, Wages, and Factor

Prices): Suppose that Assumptions 1, 2, and 3 hold, and denote the changes in pro-
ductivity, the change in aggregate output holding capital and labor constant , by ​d ln ​
Y |​K, L​​​.

17
Throughout, by “automation” or “automation technology” we refer to I​​, and use “equilibrium automation”
to refer to I​ *​.
1502 THE AMERICAN ECONOMIC REVIEW JUNE 2018

~
• If ​I * = I < ​ I​​  , so that the allocation of tasks to factors is constrained by
­technology, then _ ​​ W∗ ​ > R > _ ​ W ​​  , and
γ( ​I​​  ​) γ(N)

σ​(( γ(I ) ) ) σ​( ( γ(N) ) )

ˆ
1−​σ​ 1−​ˆ
σ​
​B​​  ​ˆσ​−1​ ​​ ​​ ____
d ln Y​|K​ , L​​ = ​ ____ ​  W*  ​ ​​​  ​ ​− R​​  1−​ˆσ​​ ​dI + _____
​ˆσ​−1
ˆ​− ​​ _
​  ​B​​   ​​​ ​R​​  1−​σ​ ​  W ​ ​​​  ​ ​dN.​
1 − ​ˆ 1 − ​ˆ
​

That is, both technologies increase productivity.

Moreover, let ​​s​L​​​ denote the share of labor in net output. The impact of t­ echnology
on factor prices in this case is given by

(σ ​ˆ + ​εL​ ​​ I )
 ​d ln W = d ln Y​|K​ , L​​ + (1 − ​sL​ ​​ )​ _____​  1  ​  ​ΛN​ ​​ dN − ​ _____ 1  ​  ​Λ​ ​​ dI ​,
​ˆ + ​εL​ ​​
​ ​
σ

( ​ˆ ​ˆσ​ + ​εL​ ​​ I )
d ln R = d ln Y​|​K, L​​ − ​sL​ ​​​ _____
​  1  ​  ​ΛN​ ​​ dN − ​ _____ 1  ​  ​Λ​ ​​ dI ​.​
σ ​+ ​εL​ ​​
 hat is, a higher N
T ​ ​ always increases the equilibrium wage but may reduce the
rental rate of capital, while a higher ​I​ always increases the rental rate of capital
but may reduce the equilibrium wage. In particular, there exists K​ ˜
​​ < ∞​ such
that an increase in ​I​increases the equilibrium wage when ​K > ​ ˜
K​​and reduces
it when ​K < ​ K˜​.
~
• If ​​I​​  ∗​ = ​ ​I < I​, so that the allocation of tasks to factors is not constrained by
technology, then ____ ​​ W* ​ = R > _ ​ W ​​  , and
γ(I ) γ(N)

σ​( ( γ(N) ) )
​ˆ 1−​ˆ
σ​
​B​​  σ​−1
d ln Y​|K​ , L​ = ​____ ​​​ ​R​​  1−​σ​ˆ​− ​​ _
​  W ​ ​​​  ​ ​dN.​
1 − ​ˆ
​


That is, new tasks increase productivity, but additional automation technologies
do not.


Moreover, the impact of technology on factor prices in this case is given by

d​ ln W = d ln ​Y |​​  K, L​​ + (1 − ​s​L​​ ) _

​  ​σ​ 1​​ + ​ε​  ​ ​​ ​ΛN​ ​​ dN,
free L

d ln R = ​d ln Y |​​  K, L​​ − ​s​L​​ _

​  ​σ​ 1​​ + ​ε​  ​ ​​ ​ΛN​ ​​ dN.​
free L

That is, an increase in ​N​(more new tasks) always increases the equilibrium wage

but may reduce the rental rate, while an increase in ​I​ (greater ­technological
automation) has no effect on factor prices.

PROOF:
See online Appendix B.
~
The most important result in Proposition 3 is that, when ​I * = I < ​ I​​, ­automation—
an increase in I​ —
​ always increases aggregate output, but has an ambiguous effect on
the equilibrium wage. On the one hand, there is a positive productivity effect ­captured
VOL. 108 NO. 6 ACEMOGLU AND RESTREPO: THE RACE BETWEEN MAN AND MACHINE 1503

by the term ​d ln Y​|K, ​ L​​​: by substituting cheaper capital for expensive labor, automa-
tion raises productivity, and hence the demand for labor in the tasks that are not yet
automated.18 Countering this, there is a negative displacement effect ­captured by
the term _____
​​σ​​ +1 ​ε​ ​​ ​  ​ΛI​ ​​​. This negative effect occurs because automation c­ ontracts the set
ˆ L
of tasks performed by labor. Because tasks are subject to ­diminishing returns in the
aggregate production function, (1), bunching workers into fewer tasks puts down-
ward pressure on the wage.
As the equation for d​ ln Y​|K​ , L​​​reveals, the productivity gains depend on the cost
savings from automation, which are given by the difference between the ­effective
wage at ​I *​, ​W/γ(I *)​ , and the rental rate, ​R​. The displacement effect dominates the
­productivity effect when the gap between W ​ /γ(I *)​and R
​ ​is small: which is guar-
anteed when K ​ < ​ ˜
K​​. In this case, the overall impact of automation on wages is
negative.
Finally, Proposition 3 shows that an increase in N ​ ​always raises productivity and
the equilibrium wage (recall that Assumption 3 imposed that ​R > W/γ(N)​). When
the productivity gains from the creation of new tasks are small, it can reduce the
rental rate of capital as well.
The fact that automation may reduce the equilibrium wage while i­ncreasing
­productivity is a key feature of the task-based framework developed here (see also
Acemoglu and Autor 2011). In our model, automation shifts the range of tasks
­performed by capital and labor: it makes the production process more c­apital
intensive and less labor intensive, and it always reduces the labor share and the
wage-rental rate ratio, ​ W/R​. This reiterates that automation is very different
from ­factor-augmenting technological changes and has dissimilar implications.
The effects of labor- or capital-augmenting technology on the labor share and the
wage-rental rate ratio depend on the elasticity of substitution (between capital and
labor). Also, capital-augmenting technological improvements always increase the
equilibrium wage, and labor-augmenting ones also do so provided that the elasticity
of substitution is greater than the share of capital in national income.19
II. Dynamics and Balanced Growth

In this section, we extend our model to a dynamic economy in which the ­evolution
of the capital stock is determined by the saving decisions of a ­representative
­household. We then investigate the conditions under which the economy admits a
balanced growth path (BGP), where aggregate output, the capital stock, and wages
grow at a constant rate. We conclude by discussing the long-run effects of a­ utomation
on wages, the labor share, and employment.

18
This discussion also clarifies that our productivity effect is similar to the productivity effect in models of
offshoring, such as Grossman and Rossi-Hansberg (2008), Rodríguez-Clare (2010), and Acemoglu, Gancia, and
Zilibotti (2015), which results from the substitution of cheap foreign labor for domestic labor in certain tasks.
19
For instance, with a constant returns to scale production function and two factors, capital and labor are q​ −​
complements. Thus, capital-augmenting technologies always increases the marginal product of labor. To see this,
let F ​ = ​FL​ ​​  , and ​​___
​ ( ​AK​ ​K, ​AL​ ​L)​be such a production function. Then W dW
d ​A​ ​​
​  = K ​FLK
​ ​ = − L ​FLL
​ ​ > 0​ (because of
K
constant returns to scale). See Acemoglu and Restrepo (2018).
Likewise, improvements in ​​AL​ ​​​increase the equilibrium wage provided that the elasticity of substitution between
capital and labor is greater than the capital share, which is a fairly weak requirement (in other words, ​​A​L​​​can reduce
the equilibrium wage only if the elasticity of substitution is low).
1504 THE AMERICAN ECONOMIC REVIEW JUNE 2018

A. Balanced Growth

We assume that the representative household’s dynamic preferences are given by

​∫0​  ​​ ​e​​  −ρt​u(C(t), L(t)) dt, ​

∞
(14) ​

where u​ (C(t), L(t))​is as defined in equation (4) and ρ

​ > 0​is the discount rate.
To ensure balanced growth, we impose more structure to the comparative
­advantage schedule. Because balanced growth is driven by technology, and in
this model sustained technological change comes from the creation of new tasks,
­constant growth requires productivity gains from new tasks to be exponential.20
Thus, in what follows we strengthen Assumption 1.

ASSUMPTION 1′: γ
​ (i)​ satisfies

γ(i) = ​e​​  Ai​ with A > 0.​

(15) ​

The path of technology, represented by ​{I(t), N(t)}​, is exogenous, and we define

n(t) = N(t) − I(t)​

​

as a summary measure of technology, and similarly let n​ * (t) = N(t) − I *(t)​be a

summary measure of the state of technology used in equilibrium (since I​ *(t) ≤ I(t)​  ,
we have n​*(t) ≥ n(t)​). New automation technologies reduce n​(t)​ , while the
­introduction of new tasks increases it.
From equation (12), aggregate output net of intermediates, or simply “net o­ utput,”
can be written as a function of technology represented by n​ *(t)​and γ
​ (I *(t)) = ​e​​  AI (t)​​,
*

the capital stock, ​K(t)​, and the level of employment, ​L(t)​, as

F(​ K(t),​  e​​  AI ​L(t); n*(t))​

*(t)
(16)  ​
ˆ
​  ​
​  σ
​
____

B​​[​(1 − n*(t))​​  ​  ​σ​ˆ ​​K​(t)​​  ​  ​ˆσ​ ​ ​]​​​ 

__ ​ˆ
σ
​−1
ˆ
ˆ
​  1  ​
∫0​ 
​  σ​
​ − ​
​+ ​​(​ )
σ
​ ​−1 ​ˆ
​
σ ____ 1
n*(t)
​​γ ​(i)​​  ​σ​ˆ−1​  di ​​​  ​ ​(​e​​  AI (t)​L(t))​​  ​ˆσ​
__ ____
=
1 *
​.​

The resource constraint of the economy then takes the form

​​K˙ ​(t) = F​(K(t), ​e​​  AI(t)​L(t); n* (t))​− C(t) − δK(t),​

where ​δ​is the depreciation rate of capital.

20
Notice also that in this dynamic economy, as in our static model, the productivity of capital is the same
in all automated tasks. This does not, however, imply that any of the previously automated tasks can be used
regardless of ​N​. As ​N​increases, as emphasized by equation (1), the set of feasible tasks shifts to the right, and
only tasks above ​N − 1​remain compatible with and can be combined with those currently in use. Just to cite a few
­motivating e­ xamples for this assumption: power looms of the eighteenth and nineteenth century are not compatible
with ­modern textile technology; first-generation calculators are not compatible with computers; many hand and
mechanical tools are not compatible with numerically controlled machinery; and bookkeeping methods from the
nineteenth and twentieth centuries are not compatible with the modern, computerized office.
 VOL. 108 NO. 6 ACEMOGLU AND RESTREPO: THE RACE BETWEEN MAN AND MACHINE 1505

We characterize the equilibrium in terms of the employment level ​L(t)​ , and the
___
normalized variables k​ (t) = K(t)​e​​  −AI (t)​​, and ​c(t) = C(t)​e​​  ​  θ ​ν(L(t))−AI (t)​.​As in our
* 1−θ *

​ (t)​denotes the rental rate, and w ​ (t) = W(t)​e​​  −AI (t)​​is the normalized
*
static model, R
wage. These normalized variables determine factor prices as

​ R(t) = ​F​K​[k(t), L(t); n*(t)]

____
​  1  ​
ˆ
​  ​ ​−1
​  σ
​−1
____ ​ˆ
σ

[ ]
__
​  1  ​
= B ​(1 − n*(t))​​  ​  ​ˆσ​ ​​ ​​ ​(1 − n *(t))​​  ​  ​σ​ˆ ​​+ ​​(​∫0​ 
( k(t) )
​ˆ
σ
​
​​ γ ​(i)​​  σ​​ˆ−1​  di)​​​  ​ ​​ _
__ __ n*(t) ˆ L(t)
​σ
​
​  ​ ​​​  ​ ​​​  ​​
1 1

and

​w(t) = F
​ ​L​​ [k(t), L(t); ​n​​  ∗​(t)]

____
​  1  ​
​  ˆ
σ
​ ​− ​
____ 1 ​ˆ
σ​−1

[
__

]
∫0
​ __ ​  1  ​

( L(t) )
1  ​
​ ​∫0​ 
​ˆ
σ​
​+ ​​(​ ​ ​di)​​​ ​ ​​​ 
ˆ​
​σ
= B​( ​di)​​​ ​​​
n*(t) ​ˆ
σ​ k(t) ​n​​ ∗​(t)
​​γ ​(i)​​  ​ˆ
σ​−1
​(1 − n*(t))​​  ​  ​ˆσ ​​​​​
__ 1
_
​  ​ ​​​ ​γ ​(i)​​ ​ˆ
σ​−1
​.​

The equilibrium interest rate is ​R(t) − δ.​

Given time paths for ​g(t)​ (the growth rate of ​​e​​  AI (t)​​) and ​n(t)​, a dynamic e­ quilibrium
*

can now be defined as a path for the threshold task ​n*(t)​, (normalized) capital and
consumption, and employment, ​{k(t), c(t), L(t)}​, that satisfies:

n*(t) ≥ n(t)​, with ​n*(t) = n(t)​only if w

• ​ ​ (t) > R(t)​, and n​ *(t) > n(t)​ only if​
w(t) = R(t)​;
• The Euler equation,
​c​(̇ t)
(17) ​​ ___ ​ = _​  1 ​ (​FK​ ​​ [k(t), L(t); n*(t)] − δ − ρ) − g(t); ​
c(t) θ
• The endogenous labor supply condition,
θ−1 ​FL​ ​​ [k(t), L(t); n*(t)]
ν ′(L(t)) ​e​​  ​  θ ​ν(L(t))​ = ​ _____________
_
(18) ​     ​ ; ​
c(t)
• The representative household’s transversality condition,

lim​​​ k(t) ​e​​  −​∫0​  ​​(​FK​ ​​[k(s), L(s); n (s)]−δ −g(s))ds​ = 0;​

t
(19) ​​ t→∞
*

• And the resource constraint,

_
(20) ​​k˙​(t) = F​(k(t), L(t); n*(t))​− c(t)​e​​  −​  θ ​ν(L(t))​− (δ + g(t))k(t).​
1−θ

We also define a balanced growth path (BGP) as a dynamic equilibrium in which

the economy grows at a constant positive rate, factor shares are constant, and the
​ (t)​is constant.
rental rate of capital R
1506 THE AMERICAN ECONOMIC REVIEW JUNE 2018

ρmin ρmax
1
Region 2A: Region 2B:
wI (n) > ρ + δ + θg > wN (n) wI (n) > ρ + δ + θg > wN (n)
n* = n n* = n

_
n(ρ)
~ n (ρ)

Region 1: Region 3:
wN (n) > ρ + δ + θg ρ + δ + θg > wI (n)
_
n* = n n* = n (ρ)

0 _ ρ
ρ

Figure 4. Behavior of Factor Prices in Different Parts of the Parameter Space

To characterize the growth dynamics implied by these equations, let us first

c­ onsider a path for technology such that ​g(t) → g​and ​n(t) → n​, consumption
grows at the rate ​g​and the Euler equation holds: ​R(t) = ρ + δ + θg​. Suppose first
that n​ *(t) = n(t) = 0​ , in which case ​F​becomes linear and ​R(t) = B​. Because the
growth rate of consumption must converge to ​g​as well, the Euler equation (17) is
satisfied in this case only if ρ
​ ​is equal to
_
​ρ
(21) ​ ​ = B − δ − θg.​

Lemma A2 in Appendix A shows that this critical value of the discount rate divides
_
the parameter space into two regions as shown in Figure 4. To the left of ​​ρ  ​​  , there
~ _ ~ _
exists a decreasing curve n​ ​​  (ρ)​defined over ​[​ρm
​ in​​, ​ρ
 ​]​ with ​​ n​(​ρ
 ​) = 0​ , and_to the right
_ _ _ _
of ​​ρ
 ​​ , there exists an increasing curve ​​  n ​(ρ)​defined over ​[​ρ  ​, ​ρm
​ ax​​]​ with ​​  n ​(​ ρ
 ​) = 0​,
such that:21
w(t)
• For ​n < ​ ~ n​(ρ)​, we have ​​_​ > R(t)​and new tasks would reduce aggregate
γ(N(t))
output, so are not adopted (recall that w ​ (t) = W(t) ​e​​  −AI (t)​​);
*

w(t)
• For ​n > ​ ~ n​(ρ)​ , we have _ ​​ ​ < R(t)​and in this case, new tasks raise
γ(N(t))
­aggregate output and are immediately produced with labor;
_
• For ​ n > ​  n ​(ρ)​ , we have ​ w(t) > R(t)​ , as a result, automated tasks raise
­aggregate output and are immediately produced with capital; and

21
The functions ​w
​ N​ ​​ (n)​and ​​wI​​​ (n)​depicted in this figure are introduced and explained below.
VOL. 108 NO. 6 ACEMOGLU AND RESTREPO: THE RACE BETWEEN MAN AND MACHINE 1507

_
• For ​n < ​  n ​(ρ)​ , we have w
​ (t) < R(t)​and additional automation would reduce
aggregate output, so small changes in automation technology do not affect ​n*​
and other equilibrium objects.

The next proposition provides the conditions under which a BGP exists, and char-
acterizes the BGP allocations in each case. In what follows, we no longer impose
Assumption 3, since depending on the value of ​ρ​ , the capital stock can become large
and violate this assumption.

PROPOSITION 4 (Dynamic Equilibrium with Exogenous Technological Change):

Suppose that Assumptions ​1′​and 2 hold. The economy admits a BGP with positive
growth if only if we are in one of the following cases:
_
(i) Full Automation: ​ρ < ​ρ  ​​ and ​ N(t) = I(t)​ (and ​ B > δ+ρ >
_
​
1 − θ ​(B − δ − ρ ) + δ​ to ensure the transversality condition). In this case,
θ
there is a unique and globally stable BGP. In this BGP, ​n* (t) = 0​ (all tasks
are produced with capital), and the labor share is zero.

(ii) Interior BGP with Immediate Automation: ​ρ ∈ ( ​ρm ​ in​, ​ρm

​ ax​)​  , ​​N˙ ​(t) = ​I˙ ​(t)
_
= Δ​  , and ​n(t) = n > max{​  n ​(ρ), ​ ​(ρ)}​
~
n (and ​ρ + (θ − 1) AΔ > 0​ to
ensure the transversality condition). In this case, there is a unique and
­globally stable BGP. In this BGP, ​n*(t) = n​ and ​I *(t) = I(t)​.
_ ˙
(iii) Interior BGP with Eventual Automation: ​ρ > ​ρ  ​​  , ​​N​(t) = Δ​ with
_
​​I˙​(t) ≥ Δ​  , and ​n(t) < ​  n ​(ρ)​ (and ​ρ + (θ − 1) AΔ > 0​ to ensure the trans-
versality ­condition). In this case, there is a unique and globally stable BGP.
_ ~
In this BGP, ​n*(t) = ​  n ​(ρ)​and ​I *(t)   = ​ ​(t)
I > I(t)​.

(iv) No Automation: ​ ρ > ​ρ​max​​​  , and ​​N˙ ​(t) = Δ​ (and ​ρ + (θ − 1) AΔ > 0​

to ensure the transversality condition). In this case, there exists a unique
and globally stable BGP. In this BGP, ​n*(t) = 1​ (all tasks are produced
with labor), and the capital share is zero.

PROOF:
See Appendix A.

The first type of BGP in Proposition 4 involves the automation of all tasks, in
which case aggregate output becomes linear in capital. This case was ruled out
by Assumption 3 in our static analysis, but as the proposition shows, when the
­discount rate, ​ρ​ , is sufficiently small, it can emerge in the dynamic model. A BGP
with no automation (case (iv)), where growth is driven entirely by the creation of
new tasks, is also possible if the discount rate is sufficiently large.
More important for our focus are the two interior BGPs where automation and
the introduction of new tasks go hand-in-hand, and as a result, n​ *(t)​is constant
at some value between 0 and 1; this implies that both capital and labor perform
a fixed measure of tasks. In the more interesting case where automated tasks are
1508 THE AMERICAN ECONOMIC REVIEW JUNE 2018

i­mmediately produced with capital (case (ii)), the proposition also highlights that
this process needs to be “balanced” itself: the two types of technologies need to
advance at exactly the same rate so that ​n(t) = n​.
Balanced growth with constant labor share emerges in this model because the
net effect of automation and the creation of new technologies proceeding at the
same rate is to augment labor while keeping constant the share of tasks performed
by labor , as shown by equation (16). In this case, the gap between the two types of
technologies, n​ (t)​ , regulates the share parameters in the resulting CES production
function, while the levels of N ​ (t)​and I​ (t)​determine the productivity of labor in the
set of tasks that it performs. When ​n(t) = n​ , technology becomes purely labor-aug-
menting on net because labor performs a fixed share of tasks and labor becomes
more productive over time in producing the newly-created tasks.22
To illustrate the main implication of the proposition, let us focus on part (ii) with
_
​​I˙​ = ​N˙​ = Δ​and ​n(t) = n ≥ ​  n ​(ρ)​. Along such a path, n​ *(t) = n​and g​ (t) = AΔ​.
Figure 5 presents the phase diagram for the system of differential equations
comprising the Euler equation (equation (17)) and the resource constraint
­
(­equation (20)). This system of differential equations determines the structure of
the dynamic equilibrium and is identical to that of the neoclassical growth model
with labor-augmenting technological change and endogenous labor supply (which
makes the locus for c​​˙​ = 0​downward-sloping because of the negative income effect
on the labor supply).

B. Long-Run Comparative Statics

We next study the log-run implications of an unanticipated and permanent decline

in ​n(t)​ , which corresponds to automation running ahead of the creation of new tasks.
Because in the short run capital is fixed, the short-run implications of this change in
technology are the same as in our static analysis in the previous section. But the fact
that capital adjusts implies different long-run dynamics.
Consider an interior BGP in which N ​ (t) − I(t) = n ∈ (0, 1)​. Along this path,
the equilibrium wage grows at the rate ​AΔ​. Define ​w ​ ​I​​ (n) = ​lim t​→∞​​ W(t)/γ(I *(t))​
as the effective wage paid in the least complex task produced with labor and​​
w​N​​ (n) = ​lim​t→∞​​ W(t)/γ(N(t))​as the effective wage paid in the most complex task
produced with labor. Both of these functions are well defined and depend only on ​n​.
Figure 4 shows how these effective wages compare to the BGP value of the rental
rate of capital, ​ρ + δ + θg​.
The next proposition characterizes the long-run impact of automation on factor
prices, employment and the labor share in the interior BGPs.

PROPOSITION 5 (Long-Run Comparative Statics): Suppose that Assumptions ​​1′​

and 2 hold. Consider a path for technology in which n​ (t) = n ∈ (0, 1)​, ​n > ​ ~
n ,
​(ρ)​
and ​g(t) = g​ (so that we are in case (ii) or (iii) in Proposition 4). In the unique
BGP we have that R ​ (t) = ρ + δ + θg​, and

22
This intuition connects Proposition 4 to Uzawa’s Theorem, which implies that balanced growth
requires a ­representation of the production function with purely labor-augmenting technological change (e.g.,
Acemoglu 2009; Grossman et al. 2017).
VOL. 108 NO. 6 ACEMOGLU AND RESTREPO: THE RACE BETWEEN MAN AND MACHINE 1509

c ċ=0

k˙ = 0

Figure 5. Dynamic Equilibrium when Technology Is Exogenous and Satisfies ​n(t) = n​

and ​g(t) = AΔ

_ _ _
• For ​n < ​  n ​(ρ)​  , we have that ​n*(t) = ​  n ​(ρ)​  , ​​wI​​​ (n) = ​wI​​​ (​  n ​(ρ))​, and ​​w​N​​ (n)
_
= ​w​N​​ (​  n ​(ρ))​. In this region, small changes in n​ ​do not affect the paths of effec-
tive wages, employment, and the labor share;
_
• For ​n > ​  n ​(ρ)​  , we have that ​n*(t) = n​  , and ​w ​ I​​​ (n)​is increasing and ​w ​ ​N​​ (n)​ is
decreasing in ​n​. Moreover, the asymptotic values for employment and the labor
share are increasing in n​ ​. Finally, if the increase in ​n​ is caused by an increase
in I​ ,​ the capital stock also increases.

PROOF:
See online Appendix B.
_
We discuss this proposition for n​ > ​  n ​(ρ)​, so that we are in the most i­nteresting
region of the parameter space where ​ I  * = I​and the level of automation is
­constrained by technology. The long-run implications of automation now differ
from its ­short-term impact. In the long run, automation reduces employment and the
labor share, but it always increases the wage. This is because in the long run capital
per worker increases to keep the rental rate constant at ​ρ + δ + θg​. This implies that
productivity gains accrue to the scarce factor, labor.23
Figure 6 illustrates the response of the economy to permanent changes in
­automation. It plots two potential paths for all endogenous variables. The dotted
line depicts the case where ​​w​I​​ (n)​is large relative to ​R​ , so that there are ­significant
productivity gains from automation. In this case, an increase in automation raises

23
This result follows because ​​wN​ ​​ (n)​is decreasing in ​n​ , and thus a lower ​n​implies a higher wage level. This
result can also be obtained by taking the log derivative of the identity (1 − η)Y = WL + RK, which implies

​d ln ​Y |​​  K, L​​ = ​sL​ ​​ d ln W + (1 − ​s​L​​ ) d ln R.​

In general, productivity gains from technological change accrue to both capital and labor. In the long run,
h­ owever, capital adjusts to keep the rental rate fixed at ​R = ρ + δ + θg​ , and as a result, ​d ln W = ​_
​sL​ ​​ ​  d ln Y ​| ​K, L​ > 0​  ,
1
meaning that productivity gains accrue only to the inelastic factor, labor.
1510 THE AMERICAN ECONOMIC REVIEW JUNE 2018

Panel A Panel B

ln W Permanent increase R
Permanent increase
in automation in automation

Initial path
for wages

ρ + δ + θA

T t T t
Panel C Panel D
Permanent increase
sL ln K Permanent increase
in automation in automation

Initial path for labor share

Initial path
for capital

T t T t

Figure 6. Dynamic Behavior of Wages (​ln W​), the Rental Rate of Capital (​R​),
the Labor Share (​​​​s​L​​​)​​​ , and the Capital Stock Following a Permanent
Increase in Automation

the wage immediately, followed by further increases in the long run. The solid
line depicts the dynamics when ​​w​I​​ (n) ≈ R​ , so that the productivity gains from
­automation are very small. In this case, an increase in automation reduces the wage
in the short run and leaves it approximately unchanged in the long run. In contrast
to the concerns that highly productive automation technologies will reduce the wage
and employment, our model shows that it is precisely when automation fails to
raise productivity significantly that it has a more detrimental impact on wages and
employment. In both cases, the duration of the period with stagnant or depressed
wages depends on θ​ ​ , which determines the speed of capital adjustment following an
increase in the rental rate.
The remaining panels of Figure 6 show that automation reduces employment
and the labor share, as stated in Proposition 5. If σ​ ˆ
​​ < 1​, the resulting capital
­accumulation mitigates the short-run decline in the labor share but does not fully
ˆ > 1​ , capital accumulation
offset it (this is the case depicted in the figure). If ​​σ​
further depresses the labor share, even though it raises the wage.
The long-run impact of a permanent increase in N ​ (t)​can also be obtained from the
proposition. In this case, new tasks increase the wage (because ​w ​ ​I​​ (n)​ is ­increasing
in n​ ​), aggregate output, employment, and the labor share, both in the short and the
long run. Because the short-run impact of new tasks on the rental rate of capital is
ambiguous, so is the response of capital accumulation.
In light of these results, the recent decline in the labor share and the ­employment
to population ratio in the United States can be interpreted as a consequence of
­automation outpacing the creation of new labor-intensive tasks. Faster automation
relative to the creation of new tasks might be driven by an acceleration in the rate
at which I​(t)​advances, in which case we would have stagnant or lower wages in
VOL. 108 NO. 6 ACEMOGLU AND RESTREPO: THE RACE BETWEEN MAN AND MACHINE 1511

the short run while capital adjusts to a new higher level. Alternatively, it might be
driven by a deceleration in the rate at which ​N(t)​advances, in which case we would
also have low growth of aggregate output and wages. We return to the p­ roductivity
implications of automation once we introduce our full model with endogenous
­technological change in the next section.

III. Full Model: Tasks and Endogenous Technologies

The previous section established, under some conditions, the existence of an

i­nterior BGP with ​​N˙ ​ = ​I˙​ = Δ​. This result raises a fundamental question: why
should these two types of technologies advance at the same rate? To answer this
question we now develop our full model, which endogenizes the pace at which
­automation and the creation of new tasks proceeds.

A. Endogenous and Directed Technological Change

To endogenize technological change, we deviate from our earlier assumption

of a perfectly competitive market for intermediates, and assume that (­intellectual)
­property rights to each intermediate, q​ (i)​, are held by a technology m ­ onopolist who
can produce it at the marginal cost μ ​ ψ​in terms of the final good, where μ ​ ∈ (0, 1)​
and ​ψ > 0​. We also assume that this technology can be copied by a fringe of
competitive firms, which can replicate any available intermediate at a higher
­
­marginal cost of ​ψ​ , and that ​μ​is such that the unconstrained monopoly price of
an intermediate is greater than ​ψ​. This ensures that the unique equilibrium price
for all types of intermediates is a limit price of ​ψ​ , and yields a per unit profit of​
(1 − μ) ψ > 0​for technology monopolists. These profits generate incentives for
creating new tasks and automation technologies.
In this section, we adopt a structure of intellectual property rights that abstracts
from the creative destruction of profits.24 We assume that developing a new
­intermediate that automates or replaces an existing task is viewed as an infringement
of the patent of the technology previously used to produce that task. For that reason,
a firm must compensate the technology monopolist who owns the property rights
over the production of the intermediate that it is replacing. We also assume that this
compensation takes place with the new inventor making a take-it-or-leave-it offer to
the holder of the existing patent.
Developing new intermediates that embody technology requires scientists.25
There is a fixed supply of S​​scientists, which will be allocated to automation
(​​S​I​​ (t) ≥ 0​) or the creation of new tasks (​​SN​ ​​ (t) ≥ 0​), so that

​S​I​​ (t) + ​S​N​​ (t) ≤ S.​

​

24
The creative destruction of profits is present in other models of quality improvements such as Aghion and
Howitt (1992) and Grossman and Helpman (1991), and will be introduced in the context of our model in Section V.
25
An innovation possibilities frontier that uses just scientists, rather than variable factors as in the ­lab-equipment
specifications (see Acemoglu 2009), is convenient because it enables us to focus on the direction of technological
change, and not on the overall amount of technological change.
1512 THE AMERICAN ECONOMIC REVIEW JUNE 2018

When a scientist is employed in automation, she automates ​κ ​ ​I​​​tasks per unit of

time and receives a wage ​​W​  SI​  ​(t)​. When she is employed in the creation of new tasks,
she creates ​​κN​ ​​​new tasks per unit of time and receives a wage ​W​ ​  SN​ ​(t)​. We assume that
automation and the creation of new tasks proceed in the order of the task index i​​.
Thus, the allocation of scientists determines the evolution of both types of
­technology, summarized by ​I(t)​and ​N(t)​  , as

(22) ​​I˙​(t) = ​κI​ ​​ ​S​I​​ (t), N˙ ​(t) = ​κN​ ​​ ​SN​ ​​ (t).​

and ​

Because we want to analyze the properties of the equilibrium locally, we make a

final assumption to ensure that the allocation of scientists varies smoothly when there
is a small difference between ​W ​ ​  SI​  ​(t)​and ​W ​ ​  SN​ ​(t)​ (rather than having d­ iscontinuous
jumps). In particular, we assume that scientists differ in the cost of effort: when
­working in automation, scientist ​j​incurs a cost of ​​χ​  Ij​ ​Y(t)​, and when working in the
­creation of new tasks, she incurs a cost of ​​χ​  Nj ​ ​Y(t)​.26 Consequently, scientist ​j​will work
​W​  SI​  ​(t) − ​W​  SN​ ​(t)
in automation if ​​____________
   ​ > ​χ​  Ij​  ​− ​χ​  Nj ​  ​​. We also assume that the ­distribution of​​
Y(t)
χ​  jI​ ​− ​χ​  jN​​​  among scientists is given by a smooth and ­increasing ­distribution ­function​
G​over a support ​[− υ, υ]​, where we take ​υ​to be small enough that ​​χ​  jI​​​  and ​​χ​  Nj ​  ​​ are
​κ​ ​​ ​V​ ​​ (t) ​κI​ ​​ ​VI​​​ (t)
{ Y(t) Y(t) }
always less than ​max​ _ ​  N N ​, _ ​  ​ ​​and thus all ­scientists always work.
​κ​ ​​
For notational convenience, we also adopt the n­ ormalization ​G(0) = _ ​​κ​ ​​ +N​κ​ ​​ ​​.
I N

B. Equilibrium with Endogenous Technological Change

We first compute the present discounted value accruing to monopolists from

automation and the creation of new tasks. Let ​V ​ ​I​​ (t)​denote the value of automating
task ​i = I(t)​ (i.e., the task with the highest index that has not yet been automated,
or more formally ​i = I(t) + ε​for ​ε​arbitrarily small and positive). Likewise, ​​V​N​​ (t)​
is the value of creating a new task at ​i = N(t)​.
To simplify the exposition, let us assume that in this equilibrium
​n(t) > max {​ n̅​(ρ), ​ ~ , so that ​I *(t) = I(t)​and newly-automated tasks start being
​(ρ)}​
n
produced with capital immediately. The flow profits that accrue to the technology
monopolist that automated task i​​ are

​πI​ ​​ (t, i ) = bY(t) R​(t)​​  ζ−​ˆσ​​, ​

(23) ​

where ​b = (1 − μ)​B​​  ​ˆσ​−1​  η ​ψ​​  1−ζ​​.27 Likewise, the flow profits that accrue to the
­technology monopolist that created the labor-intensive task ​i​ are
ˆ
ζ−​σ​

( γ(i) )
W(t)
​πN​ ​​ (t, i ) = bY(t) ​​ _
(24) ​ ​  ​ ​​​  ​.​

26
The cost of effort is multiplied by ​Y(t)​to capture the income effect on the costs of effort in a tractable manner.
27
This expression follows because the demand for intermediates is ​ q(i) = ​ B​​  σ​​ˆ−1​  η ​ψ​​  −ζ​Y(t )R​(t )​​  ζ−​σ​
ˆ​​,

every i­ntermediate is priced at ​ψ​and the technology monopolist makes a per unit profit of ​1 − μ​.
VOL. 108 NO. 6 ACEMOGLU AND RESTREPO: THE RACE BETWEEN MAN AND MACHINE 1513

The take-it-or-leave-it nature of offers implies that a firm that automates task I​​
needs to compensate the existing technology monopolist by paying her the present
discounted value of the profits that her inferior labor-intensive technology would
generate if not replaced. This take-it-or-leave-it offer is given by28

ˆ
ζ−​σ​
∫t −​∫0​  ​​(R(s)−δ)ds
( γ(I ) )
W(τ) ∞ τ

b ​ ​  ​​e​​ 
​ ​Y(τ) ​​ _
​  ​ ​​​  ​dτ.​

Likewise, a firm that creates task ​N​needs to compensate the existing ­technology
monopolist by paying her the present discounted value of the profits from the
­capital-intensive alternative technology. This take-it-or-leave-it offer is given by

b ​∫t​  ​​ ​e​​  −​∫0​  ​​(R(s)−δ)ds​Y(τ ) R ​(τ )​​  ζ−​σ​

∞ τ

​ ˆ​dτ.​

In both cases, the patent-holders will immediately accept theses offers and reject
less generous ones.
We can then compute the values of a new automation technology and a new task,
respectively, as

​​(R ​(τ)​​  ζ−​ˆσ​​− ​​(w(τ) ​e​​  ​∫t​  ​​g(s)ds​)​​​  ​)​dτ, ​

ζ−​ˆ
σ​
(25) ​​V​I​(t) = bY(t)​∫t​  ​​e​​  −​∫t​ 
∞   τ τ
​​(R(s)−δ−​gy​​​(s))ds

and

(( γ(n(t)) ​​e​​  )
ˆ
ζ−​σ​
(26) ​​VN​ ​(t) = bY(t)​∫t​  ​​e​​  −​∫t​  ​​(R(s)−δ−​gy​​​(s))ds​​ ​​ _
)
∞ τ
w(τ) ​∫​  τ​​g(s)ds
​  t ​ ​​​  ​− R ​(τ )​​  ζ−​ˆσ​​ ​dτ,​

where ​​gy​​(t)​is the growth rate of aggregate output at time t​​and as noted above, g​ (t)​
is the growth rate of γ ​ (N(t))​.
To ensure that these value functions are well behaved and non-negative, we impose
the following assumption for the rest of the paper.

ˆ > ζ​.
ASSUMPTION 4: ​​σ​

This assumption ensures that innovations are directed toward ­ technologies

that allow firms to produce tasks by using the cheaper (or more productive)
­factors and, consequently, that the present discounted values from innovation are
­positive. This assumption is intuitive and reasonable: since intermediates embody
the t­echnology that directly works with labor or capital, they should be highly
­complementary with the relevant factor of production in the production of tasks.29

28
This expression is written by assuming that the patent-holder will also turn down subsequent less generous
offers in the future. Deriving it using dynamic programming and the one-step-ahead deviation principle leads to
the same conclusion.
29
The profitability of introducing an intermediate that embodies a new technology depends on its demand.
As a factor (labor or capital) becomes cheaper, there are two effects on the demand for q​ (i )​. First, the decline in
costs allows firms to scale up their production, which increases the demand for the intermediate good. The extent
1514 THE AMERICAN ECONOMIC REVIEW JUNE 2018

The expressions for the value functions, ​V ​ ​I​​ (t)​and ​V

​ ​N​​ (t)​in equations (25) and (26)
are intuitive. The value of developing new automation technologies depends on the
gap between the cost of producing with labor (given by the effective wage, ​w(τ)​)
and the rental rate of capital (recall that ​​σ​ ˆ > ζ​). When the wage is higher, ​​VI​​​ (t)​
increases and technology monopolists have greater incentives to introduce new
automation technologies to substitute capital for the more expensive labor.
­
The ­expression for ​V ​ ​N​​ (t)​has an analogous interpretation, and is greater when the
gap between the rental rate of capital and the cost of producing new tasks with labor
(​w(τ )/γ(n(t ))​) is larger.30
An equilibrium with endogenous technology is given by paths {​ K(t ), N(t ), I(t )}​
for capital and technology (starting from initial values K ​ (0), N(0), I(0)​), paths​
{ R(t ), W(t ), ​W​  SI​  ​(t ), ​W​  SN​ ​(t )}​for factor prices, paths ​ { ​VN​ ​​ (t ), ​VI​​​ (t )}​for the value
­functions of technology monopolists, and paths ​{ ​SN​ ​​ (t ), ​SI​​​ (t )}​for the allocation of
scientists such that all markets clear, all firms and prospective technology monopo-
lists maximize profits, and the representative household maximizes its utility. Using
the same normalizations as in the previous section, we can represent the equilibrium
with endogenous technology by a path of the tuple {​ c(t ), k(t ), n(t ), L(t ), ​SI​​​ (t ), ​SN​ ​​ (t ),​
V​I​​ (t ), ​VN​ ​​ (t )}​such that

• Consumption satisfies the Euler equation (17) and the labor supply satisfies
equation (18);
• The transversality condition holds

lim​​(k(t ) + Π(t )) ​e​​  −​∫0​  ​​(ρ−(1−θ)g(s))ds​ = 0,​

t
(27) ​​t→∞

where in addition to the capital stock, the present value of corporate profits​
Π(t ) = I(t) ​VI​​​ (t )/Y(t ) + N(t ) ​VN​ ​​ (t )/Y(t)​is also part of the representative
household’s assets;
• Capital satisfies the resource constraint
η
[ ]
​​k ​(̇ t ) = ​ 1 + _
_
​   ​ (1 − μ) ​F(​ k(t), L(t); n*(t))​− c(t) ​e​​  − ​  θ ​ν(L(t))​− (δ + g(t)) k(t),​
1−θ
1−η

of this positive scale effect is regulated by the elasticity of substitution ˆ σ​

​​ ​. Second, because the cheaper factor
is substituted for the intermediate with which it is combined, the demand for that intermediate good falls. This
­countervailing s­ ubstitution effect is regulated by the elasticity of substitution ​ζ​. The condition ​​ˆ
σ​ > ζ​ guarantees
that the former, positive effect dominates, so that prospective technology monopolists have an incentive to introduce
­technologies that allow firms to produce tasks with cheaper factors. When the opposite holds, i.e., ζ​ > ​ˆ σ​​ , we have
the p­ aradoxical situation where technologies that work with more expensive factors are more profitable. In this case,
the present discounted values from innovation are negative.
30
There is an important difference between the value functions in (25) and (26) and those in models of directed
technological change building on factor-augmenting technologies (such as in Acemoglu 1998, 2002). In the latter
case, the direction of technological change is determined by the interplay of a market size effect favoring the more
abundant factor and a price effect favoring the cheaper factor. The task-based framework here, combined with the
assumption on the structure of patents, makes the benefits of new technologies only a function of the factor prices:
in particular, the difference between the wage rate and the rental rate. This is because factor prices determine the
profitability of producing with capital relative to labor. Without technological constraints, this would determine the
set of tasks that the two factors perform. In the presence of technological constraints restricting which tasks can
be produced with which factor, factor prices determine the incentives for automation (to expand the set of tasks
produced by capital) and the creation of new tasks (to expand the set of tasks produced by labor).
We should also note that despite this difference, the general results on absolute weak bias of technology in
Acemoglu (2007) continue to hold here, in the sense that an increase in the abundance of a factor always makes
technology more biased toward that factor.
VOL. 108 NO. 6 ACEMOGLU AND RESTREPO: THE RACE BETWEEN MAN AND MACHINE 1515

where recall that F ​ ​(k(t ), L(t ); n*(t ))​​is net output (aggregate output net of
η
­intermediates) and ​​_​ (1 − μ ) F​(k(t ), L(t ); n*(t ))​​is profits of technology
1−η
monopolists from intermediates;
• Competition among prospective technology monopolists to hire scientists
​ ​  SI​  ​(t ) = ​κI​ ​​ ​VI​​​ (t )​and ​​W​  SN​ ​(t ) = ​κN​ ​​ ​VN​ ​​ (t)​. Thus,
implies that ​W

[ Y(t ) )]
​κ​ ​​ ​V​​​ (t ) ​κN​ ​​ ​VN​ ​​ (t ) ​κI​ ​​ ​VI​​​ (t ) _ ​κ​ ​​ ​V​ ​​ (t )
( Y(t ) ) (
​​SI​​​ (t ) = SG​ _
​  I I ​  − _ ​  ​ ​, ​S​N​​ (t ) = S​ 1 − G​ ​  _ ​− ​  N N ​ ​ ​,​
Y(t ) Y(t )

and ​n(t )​evolves according to the differential equation,

​κ​ ​​ ​V​​​ (t ) ​κN​ ​​ ​VN​ ​​ (t )
( Y(t ) Y(t ) )
(28) ​​ṅ ​(t ) = ​κN​ ​​ S − (​κN​ ​​ + ​κI​ ​​ ) G​ _
​  I I ​− _ ​  ​ ​S;​

​ I​​​ (t )​ and​​
• And the value functions that determine the allocation of scientists, ​V
V​N​​ (t )​ , are given by (25) and (26).

As before, a BGP is given by an equilibrium in which the normalized variables​

c(t ), k(t )​, and L ​ (t )​, and the rental rate R ​ (t )​are constant, except that now ​n(t )​ is
­determined endogenously. The definition of the equilibrium shows that the p­ rofits
from automation and the creation of new tasks determine the evolution of n​ (t )​:
whenever one of the two types of innovation is more profitable, more scientists will
be allocated to that activity.
Consider an allocation where n​ (t ) = n ∈ (0, 1)​. Let us define the n­ ormalized
value functions ​ ​vI​​​ (n) = ​lim​t→∞​​ ​VI​​​ (t )/Y(t)​and ​ ​vN​ ​​ (n) = ​lim​t→∞​​ ​VN​ ​​ (t )/Y(t)​,
which only depend on ​n​. Equation (28) implies that ​n​̇ ​(t ) > 0​if and only if
​​κN​ ​​ ​VN​ ​​ (t ) > ​κI​ ​​ ​VI​​​ (t)​, and ​​ṅ ​(t ) < 0​if and only if ​​κN​ ​​ ​VN​ ​​ (t ) < ​κI​ ​​ ​VI​​​ (t )​. Thus, if
​​κI​ ​​ ​vI​​​ (n) ≠ ​κN​ ​​ ​vN​ ​​ (n)​ , the economy converges to a corner with ​n(t)​equal to 0 or 1,
and for an interior BGP with n​ ∈ (0, 1)​we need

​κI​ ​​ ​vI​​​ (n) = ​κN​ ​​ ​vN​ ​​ (n).​

(29) ​

The next proposition gives the main result of the paper, and characterizes ­different
types of BGPs with endogenous technology.

PROPOSITION 6 (Equilibrium with Endogenous _Technological Change): _

Suppose that Assumptions 1​ ′​, 2, and 4 hold. There exists ​ ​S ​​such that, when ​S < ​  S ​​,
we have:31
_
(i) Full Automation: For ρ ​ < ​ρ ​​  , there is a BGP in which n​ (t ) = 0​
and thus all tasks are produced with capital (this case also requires​
B > δ + ρ > ​ 1 − θ ​(B − δ − ρ) + δ​ to ensure the transversality condition).
____
θ

_
31
The condition S​ < _​  S ​​ensures that the growth rate of the economy is not too high. If the growth rate is above
the threshold implied by ​ ​S ​​ , the creation of new tasks is discouraged (even if current wages are low) because firms
anticipate that the wage will grow rapidly, reducing the future profitability of creating new labor-intensive tasks.
This condition also allows us to use Taylor approximations of the value functions in our analysis of local stability.
Finally, in parts (ii)–( iv) this condition ensures that the transversality condition holds.
1516 THE AMERICAN ECONOMIC REVIEW JUNE 2018

_ _ _
For ​ρ > ​ρ​​  , all BGPs feature ​n(t ) = n > ​n ​(ρ)​. Moreover, there exists ​​κ
  ​ ≥ ​
κ
_ ​ > 0​ such that:
_
(ii) Unique Interior BGP: if ​κ ​ ​I​​/​κN​ ​​  > ​κ
 ​​ there exists a unique BGP. In this
_
BGP we have n​ *(t ) = n(t ) = n ∈ (​  n ​( ρ), 1)​ and ​​κN​ ​​ ​vN​ ​​ (n) = ​κI​ ​​ ​vI​​​ (n)​. If,
in addition, ​θ = 0​  , then the equilibrium is unique everywhere and the BGP
is globally (saddle-path) stable. If θ​ > 0​  , then the equilibrium is unique in
the neighborhood of the BGP and is asymptotically (saddle-path) stable;
_ ​κI​ ​​
(iii) Multiple BGPs: if ​​κ ​ > _ ​ N​  ​ ​​ > ​
​  κ κ ​​  , there are multiple BGPs;
_
​κ​ ​​
(iv) No Automation: If ​​κ _​ > _ ​  ​κ​ I  ​​ ​​ , there exists a unique BGP. In this BGP​
n*(t ) = 1​ and all tasks are produced with labor. (When ​ρ > ​ρ​max​​​  , we are
N

always in this case.)

PROOF:
See Appendix A.

This proposition provides a complete characterization of different types of BGPs.

Figure 7 shows visually how different BGPs arise in parts of the parameter space.
Further intuition can be gained by studying the behavior ​​κI​ ​​ ​vI​​​ (n)​and ​​κ​N​​ ​vN​ ​​ (n)​  ,
which we do in Figure 8. Lemma A3 shows that, for ​S​small, the normalized value
functions can be written as
b​(​( ρ + δ + θg)​​  ζ−​ˆσ​​− ​wI​​​(n)​​  ζ−​ˆσ​​)​
​​v​I​(n) = ​ _____________________
        ​ ,
ρ + (θ − 1)g

b​(​​wN​ ​​(n)​​  ζ−​ˆσ​​− ​( ρ + δ + θg)​​  ζ−​ˆσ​​)​

v​N​(n) = ​ _____________________
​         ​ ​.
ρ + (θ − 1)g

The profitability of the two types of technologies depends on the effective wages,​​
w​I​​ (n)​and ​​w​N​​ (n)​. A lower value of ​n​ , which corresponds to additional ­automation,
reduces ​​wI​​​ (n)​: in other words ​​wI​​​ (n)​is increasing in n​ ​. This is because of c­ omparative
advantage: as more tasks are automated, the equilibrium wage increases less than​
γ(I)​ , and it becomes cheaper to produce the least complex tasks with labor, and thus
automation becomes less profitable. Because ​w ​ ​I​​ (n)​is increasing in n​ ​ , so is ​v​ I​​​ (n)​
(recall that ​σ ​ˆ ​ > ζ​). However, ​v​ ​N​​ (n)​is also increasing in n​ ​: ​​wN​ ​​ (n)​is decreasing in n​ ​
as the long-run wage increases with automation owing to the productivity effect dis-
cussed in the previous section. We will see next that the fact that ​v​ N​ ​​ (n)​is increasing
in n​ ​creates a force toward multiplicity of BGPs, while the fact that ​v​ I​​​ (n)​is increas-
ing in ​n​pushes toward uniqueness and stability.
Panel A of Figure 8 illustrates the first part of Proposition 6 (which parallels the
_
first part of Proposition 4): when ​ρ < ​ρ  ​​  , ​​κI​ ​​ ​vI​​​ (0)​is above ​​κN​ ​​ ​vN​ ​​ (0)​for ​n < ​ ~
n ρ)​.
​(
In this region it is not optimal to create new tasks. Consequently, there exists a BGP
with full automation, meaning that all tasks will be automated and produced with
capital. Reminiscent of Leontief’s “horse equilibrium,” in this BGP labor becomes
_
redundant. Intuitively, as also shown in Figure 4, when ​ρ < ​ρ  ​​and ​n < ​ ~ n , we
​(ρ)​ 
VOL. 108 NO. 6 ACEMOGLU AND RESTREPO: THE RACE BETWEEN MAN AND MACHINE 1517

κI
κN
___

Unique interior BGP

κ
Full _
automation
BGP
Multiple BGP κ
_

Unique BGP
with no automation

_ ρmax ρ
ρ

Figure 7. Varieties of BGPs

Panel A Panel B
κ I′ v I (n) κI v I (n)

κN v N (n)
κI v I (n)
κN v N (n)

_
n(ρ)
~ n n (ρ) n
_ _
ρ< ρ ρ> ρ

Figure 8. Asymptotic Behavior of Normalized Values

Notes: Asymptotic behavior of normalized values. In panel A, ​​​​κI​​ ​​​​​ ​​​vI​​​ ​​​​​  (n)​ is everywhere ​​​​κ​N​​ ​​​​​​​​v​N​​ ​​​​​  (n)​ and the BGP involves
full automation. In panel B, if ​​​​​​κI​​ ​​​/​​​κ​N​​ ​​​​​​​​is sufficiently large, the two curves intersect, and we have an interior BGP
with both automation and c­ reation of new tasks. Panel B also shows the effect of an increase in the productivity of
­scientists in automating tasks from ​​​​κ​​I​​​​​​​​ to ​​​​​​κ​​  ′I​​.​ 

have ​​wN​ ​​ (n) > ρ + δ + θg​ , which implies that labor is too expensive relative to
­capital. Utilizing and thus creating new tasks is not profitable. Economic growth in
this BGP is driven by capital accumulation (because when all tasks are automated,
aggregate output is linear in capital).
Panel B illustrates the remaining three types of BGPs, which apply when ​ρ >​
_
 ​​. In this case, at ​n = 0​ (or at any ​n ≤ ​ n̅​( ρ)​), ​κ
ρ ​ ​I​​ ​vI​​​ (n)​is strictly below ​​κN​ ​​ ​vN​ ​​ (n)​
and thus a full automation BGP is not possible. The two curves can only inter-
sect for n​ ∈ (​ n̅​( ρ), 1]​ , implying that in any BGP, newly automated tasks will be
immediately produced with capital. As explained above, both of these curves are
increasing but their relative slopes depend on ​​κ​I​​/​κN​ ​​​. When ​​κI​ ​​/​κN​ ​​  < ​κ _​​, ​​κ​I​​ ​vI​​​ (n)​
1518 THE AMERICAN ECONOMIC REVIEW JUNE 2018

is not ­sufficiently steep relative to ​​κN​ ​​ ​vN​ ​​ (n)​, and the two never intersect. This
means that even at ​n = 1​, it is not profitable to create new automation technolo-
gies, and all tasks will be produced with labor. In this BGP, capital becomes redun-
dant, and growth is driven by endogenous technological change increasing labor’s
­productivity as in the standard quality ladder models such as Aghion and Howitt
(1992) or Grossman and Helpman (1991).
_
Conversely, when κ​ ​​ I​​/​κN​ ​​ > ​κ ​​ , the curve ​κ​ I​ ​​ ​vI​​​ (n)​is sufficiently steep relative to
​​κN​ ​​ ​vN​ ​​ (n)​so that the two curves necessarily intersect and can only intersect once.
Hence, there exists a unique interior BGP (interior in the sense that now the BGP
level of ​n​is strictly between 0 and 1, and thus some tasks are produced with labor
and some with capital).
_
Finally, when ​​κ  ​ > ​κI​ ​​/​κN​ ​​  > ​κ _​​ , the two curves will intersect, but will do so
­multiple times, leading to multiple interior BGPs.
_
Proposition 6 also shows that for ​​κ​I​​/​κN​ ​​  > ​κ  ​​ , the unique interior BGP is globally
­stable provided that the intertemporal elasticity of substitution is infinite (i.e., ​θ = 0​),
and locally stable otherwise (i.e., when θ​ > 0​). Because ​κ ​ ​I​​ ​vI​​​ (n)​starts below
​​κ​N​​ ​vN​ ​​ (n)​at ​ ​n̅​(ρ)​ (reflecting the fact that at this point, new automation t­echnologies
are not immediately adopted and thus the value of creating these technologies is 0),
the unique intersection must have the former curve being steeper than the former. At
this point, a further increase in ​n​always raises the value of automating an additional
task, ​​vI​​​ (n)​, more than the value of creating a new task, ​v​ N​ ​​ (n)​. This ensures that
increases in ​n​beyond its BGP value trigger further automation, while lower values
of n​ ​encourage the creation of new tasks, ensuring the stability of the unique BGP.
The asymptotic stability of the interior BGP implies that there are ­powerful ­market
forces pushing the economy toward balanced growth. An important c­ onsequence
of this stability is that technological shocks that reduce n​ ​ (e.g., the arrival of a
series of new automation technologies) will set in motion self-correcting forces.
Following such a change, there will be an adjustment process restoring the level of
­employment and the labor share back to their initial values.
This does not, however, imply that all shocks will leave the long-run prospects
of labor unchanged. For one, this would not necessarily be the case in a situation
with multiple steady states, and moreover, certain changes in the environment
(for ­example, a large increase in ​B​or a decline in ​ρ)​ can shift the economy from the
region in which there is a unique interior BGP to the region with full ­automation,
with disastrous consequences for labor. In addition, the next corollary shows that,
if there is a change in the innovation possibilities frontier (in the κ ​ ​s) that makes
it permanently easier to develop new automation technologies, self-correcting
forces still operate but will now only move the economy to a new BGP with lower
­employment and a lower labor share.
_ _
COROLLARY 2: Suppose that ρ ​ > ​ρ  ​​ and ​​κI​ ​​/​κN​ ​​  > ​κ
 ​​. A one-time permanent
​ I​ ​​/​κN​ ​​​leads to a BGP with lower ​n​, employment and labor share.
increase in ​κ

This corollary follows by noting that an increase in ​κ ​ ​I​​/​κN​ ​​​shifts the intersection
​ I​ ​​ ​vI​​​ (n)​and ​κ
of the curves ​κ ​ ​N​​ v(n)​to the left as shown by the blue dotted curve in
Figure 8, leading to a lower value of n​ ​in the BGP. This triggers an adjustment
­process in which the labor share and employment decline over time, but ultimately
VOL. 108 NO. 6 ACEMOGLU AND RESTREPO: THE RACE BETWEEN MAN AND MACHINE 1519

settle to their new (interior) BGP values. The transition process will involve a
slower rate of increase of N ​ ​and a more rapid rate of increase of I​​than the BGP.
Interestingly, if new tasks generate larger productivity gains than automation, this
transition process will also be associated with a slowdown in productivity growth
because automation crowds out resources that could be used to develop new tasks.32
In summary, Proposition 6 characterizes the different types of BGPs, and together
with Corollary 2, it delineates the types of changes in technology that ­trigger
­self-correcting dynamics. Starting from the interior BGP, the effects of (small)
increases in automation technology will reverse themselves over time, restoring
employment and the labor share back to their initial values. Permanent changes in
the ability of society to create new automation technologies trigger self-correcting
dynamics as well, but these will take us toward a new BGP with lower employment
and labor share, and may also involve slower productivity growth in the process.

IV. Extensions

In this section we discuss three extensions. First we introduce heterogeneous

skills, which allow us to analyze the impact of technological changes on inequality.
Second, we study a different structure of intellectual property rights that introduces
the creative destruction of profits. Finally, we discuss the welfare implications of
our model.

A. Automation, New Tasks, and Inequality

To study how automation and the creation of new tasks impact inequality, we now
introduce heterogeneous skills. This extension is motivated by the observation that
both automation and new tasks could increase inequality: new tasks favor ­high-skill
workers who tend to have a comparative advantage in new and complex tasks,
while automation substitutes capital for labor in lower-indexed tasks where l­ ow-skill
workers have their comparative advantage.
The assumption that high-skill workers have a comparative advantage in new
tasks receives support from the data. Figure 9 shows that occupations with more new
job titles in 1980, 1990, and 2000 employed workers with greater average years of
schooling.33
To incorporate this feature, we assume that there are two types of workers:
­low-skill workers with time-varying productivity ​​γL​ ​​ (i, t)​in task ​i​ , and high-skill
​ ​​ (i)​. We parametrize these productivities as follows.
workers with productivity ​​γH

32
Forgone productivity gains from slower creation of new tasks will exceed the gains from automation, causing
a productivity slowdown during a transition to a higher level of automation, if ​ρ > ​ρ​P​​​ , where ​​ρP​ ​​​is defined implic-
itly as the solution to the equation
​​____ ____
1 ​  (​w​​​​(n)​​  1−σ​− ​( ​ρ​ ​​ + δ + θg)​​  1−σ​) = ​ 1 ​  (​( ​ρ​ ​​ + δ + θg)​​  1−σ​− ​w​ ​​​(n)​​  1−σ​).​ 
σ−1 I P
σ−1 P N

33
As in Figure 1, this figure partials out the demographic composition of employment in each occupation at the
beginning of the relevant period. See online Appendix B for the same relationship without these controls as well as
with additional controls.
1520 THE AMERICAN ECONOMIC REVIEW JUNE 2018

18
1980
1990
2000
16
Average years of schooling

14

12

10

0 0.2 0.4 0.6 0.8

Share of new job titles in each decade

Figure 9. Average Years of Schooling among Workers and the Share of New Job
Titles in 1980, 1990, and 2000

Note: See online Appendix B for data sources and detailed definitions.

ASSUMPTION ​1″​: The productivities of high-skill and low-skill workers are

given by

​ ​​(i ) = e​ ​​  ​AH​ ​​i​, ​γ​L​​ (i, t ) = e​ ​​  ξ​AH​ ​​i​Γ(t − T(i)),​

​​γH

where ​Γ​ is increasing with ​​lim​x→∞​​ Γ(x) = 1​, ​ξ ∈ (0, 1]​, and ​T(i)​ denotes the time
when task i​​was first introduced.

Assumption ​1″​is similar to but extends Assumption ​1′​in several d­ imensions.

The ratio ​​γ​H​​ (i)/​γ​L​​ (i, t)​is increasing in ​i​, which implies that high-skill workers
have a c­ omparative advantage in higher-indexed tasks. But in addition, we also
let the p­ roductivity of low-skill workers in a task increase over time, as captured
by the increasing function ​Γ​. This captures the idea that as new tasks become
“­standardized,” they can be more productively performed by less skilled w ­ orkers
(e.g., Acemoglu, Gancia, and Zilibotti 2012), or that workers adapt to new
­technologies by a­ cquiring human c­ apital through training, on-the-job learning, and
schooling (e.g., Schultz 1975; Nelson and Phelps 1966; Galor and Moav 2000;
Goldin and Katz 2008; and Beaudy, Green, and Sand 2016). Since the function ​Γ​
limits to 1 over time, the parameter ​ξ​determines whether this standardization effect
is complete or i­ncomplete. When ​ξ < 1​, the productivity of low-skill w ­ orkers
­relative to ­high-skill ­workers ­converges to ​​γL​ ​​ (i, t)/​γH
​ ​​ (i) = ​γH
​ ​​ ​(i)​​  ξ−1​​, and limits to
zero as more and more advanced tasks are introduced. In contrast, when ξ​ = 1​, the
relative productivity of low-skill ­workers converges to 1​ ​for tasks that have been
around for a long time.
VOL. 108 NO. 6 ACEMOGLU AND RESTREPO: THE RACE BETWEEN MAN AND MACHINE 1521

The structure of comparative advantage ensures that there exists a threshold task​
M​such that high-skill labor performs tasks in ​​[M, N]​, low-skill labor performs tasks
in (​ I *, M )​, and capital performs tasks in [​​N − 1, I *]​. In what follows, we denote the
wages of high and low-skill labor by ​W ​ ​​and ​W
​ H ​ ​L​​, respectively, and to simplify the
discussion, we focus on the economy with exogenous technology and assume that the
supply of high-skill labor is fixed at ​H​and the supply of low-skill labor is fixed at L ​ ​  .

PROPOSITION 7 (Automation, New Tasks, and Inequality): Suppose Assumptions​

1″​ and 2 hold. Suppose also that technology evolves exogenously with N ​​˙ ​ = ​I˙​ = Δ​
_
and ​n(t) = n > max{​  n ​( ρ), ​ ​(
n ρ)}​ (and ​​AH
~ ​ ​​ (1 − θ )Δ < ρ​). Then, there exists a
unique BGP. Depending on the value of ​ξ​this BGP takes one of the following forms:

If ​ξ < 1​  , in the unique BGP we have ​l​im​t→∞​​ ​WH

(i)  ​ ​​ (t )/​WL​ ​​ (t ) = ∞​, the share
of tasks performed by low-skill workers converges to zero, and capital and
high-skill workers perform constant shares of tasks.

(ii) If ​ξ = 1​  , in the unique BGP ​W ​ H ​ ​​ (t)​ and ​​WL​ ​​ (t)​ grow at the same rate as
the economy, the wage gap, ​​WH ​ ​​ (t )/​WL​ ​​ (t )​, remains constant, and capital,
low-skill, and high-skill workers perform constant shares of tasks. Moreover,​​
lim​t→∞​​ ​WH​ ​​ (t )/​WL​ ​​ (t)​is decreasing in ​n​. Consequently, a permanent increase
in ​N​raises the wage gap ​​W​H​​ (t )/​WL​ ​​ (t)​in the short run, but reduces it in the
long run, while a permanent increase in I​​ raises the wage gap in both the
short and the long run.

Like all remaining proofs in the paper, the proof of this proposition is in online
Appendix B.
When ​ ξ < 1​ , this extension confirms the pessimistic scenario about the
­implications of new technologies for wage inequality and the employment ­prospects
of low-skill workers : both automation and the creation of new tasks increase inequal-
ity, the former because it displaces low-skill workers ahead of h­ igh-skill workers,
and the latter because it directly benefits high-skill workers who have a comparative
advantage in newer, more complex tasks relative to low-skill workers. As a result,
low-skill workers are progressively squeezed into a smaller and smaller set of tasks,
and wage inequality grows without bound.
However, our extended model also identifies a countervailing force, which becomes
particularly potent when ξ​ = 1​. Because new tasks become standardized, they can
over time be as productively used by low-skill workers. In this case, automation
and the creation of new tasks still reduce the relative earnings of low-skill workers
in the short run, but their long-run implications are very different. In the long run,
­inequality is decreasing in ​n​ (because a higher ​n​translates into a greater range of
tasks for low-skill workers). Consequently, automation increases inequality both in
the short and the long run. The creation of new tasks, which leads to a permanently
higher level of ​n​, increases inequality in the short run but reduces it in the long
run. These observations suggest that inequality may be high following a period of
adjustment in which the labor share first declines (due to increases in automation),
and then recovers (due to the introduction and later standardization of new tasks).
1522 THE AMERICAN ECONOMIC REVIEW JUNE 2018

B. Creative Destruction of Profits

In this subsection, we modify our baseline assumption on intellectual p­ roperty

rights and revert to the classical setup in the literature in which new t­ echnologies do
not infringe the patents of the products that they replace (Aghion and Howitt 1992;
Grossman and Helpman 1991). This assumption introduces the c­ reative d­ estruction
effects: the destruction of profits of previous inventors by new ­innovators. We will see
that this alternative structure has similar implications for the BGP, but n­ ecessitates
more demanding conditions to guarantee its u­ niqueness and stability.
Let us first define ​​V​N​​ (t, i)​and ​​VI​​​ (t, i)​as the time ​t​values for technology ­monopolist
with, respectively, new task and automation technologies. These value functions
­satisfy the following Bellman equations:

r (t) ​VN​ ​​ (t, i ) − ​​V ​​Ṅ ​​ (t, i ) = ​πN​ ​​ (t, i ),

​ r (t) ​VI​​​ (t, i ) − ​​V ​​İ ​​ (t, i ) = ​πI​ ​​ (t, i ).​

Here ​π
​ I​ ​​ (t, i )​and ​​πN​ ​​ (t, i )​denote the flow profits from automating and creating new
tasks, respectively, which are given by the formulas in equations (23) and (24).
For a firm creating a new task i​ ​, let ​​T​​  N​(i)​denote the time at which it will be replaced
by a technology allowing the automation of this task. Likewise, let ​T ​ ​​  I​(i)​ denote
the time at which an automated task ​i​will be replaced by a new task using labor.
Since firms anticipate these deterministic replacement dates, their value f­unctions
also satisfy the boundary conditions ​ ​VN​ ​​ (​T​​    N​(i ), i ) = 0​and ​
​VI​​​ (​T​​  I​(i ), i ) = 0​.
Together with these boundary conditions, the Bellman equations solve for
ˆ
ζ−​σ​
= ​VN​ ​​ (N(t ), t ) = b​∫​ 
W(τ )
( γ(N(t )) )
​T​​  N​(N(t)) −​∫​  ​​(R(s)−δ)ds
τ

​​V​  CD
N​  ​(t ) ​​ ​e​​  t ​Y(τ) ​​ _
​  ​ ​​​  ​dτ,
t
ζ−​ˆ
​
σ
W(τ )
( γ(I(t )) })
= ​VI​​​ (I(t ), t ) = b​∫​ 
{
​T​​  I​(I(t)) −​∫​  ​​(R(s)−δ)ds
τ

I​  ​(t )
​V​  CD ​​ ​e​​  t ​Y(τ )​​ min​ R(τ ), _
​  ​ ​ ​​ ​ ​dτ.​
t

For reasons that will become evident, we modify the innovation possibilities
frontier to

(30) ​​I˙​(t ) = ​κ​I​​ ι(n(t )) ​SI​​​ (t ), N˙ ​(t ) = ​κN​ ​​ ​S​N​​ (t )​.

and ​

Here, the function ​ι(n(t ))​is included and assumed to be nondecreasing to capture the
possibility that automating tasks closer to the frontier (defined as the h­ ighest-indexed
task available) may be more difficult.
Let us again define the normalized value functions as ​ I​  ​  (n)
​v​  CD
​_
V​  I​  ​(t)
CD
​
V
_ ​  CD
​  ​
(t)
= ​lim​t→∞​​ N​  ​(n) = ​lim​t→∞​​
​​and ​​v​  CD N
​​ . In a BGP, the normalized value
Y(t) Y(t)
­functions only depend on ​n​because newly-created tasks are automated after a
period of length ​​T​​  N​(N(t)) − t = n/Δ​ , and newly-automated tasks are replaced by
​κ​ ​​ ​κ​ ​​ ι(n)
new ones after a period of length ​T ​ ​​  I​(I(t)) − t = _ ​1 − n ​​ , where Δ
​ = _​ I N ​  S​ is
Δ ​κI​ ​​ ι(n) + ​κN​ ​​
VOL. 108 NO. 6 ACEMOGLU AND RESTREPO: THE RACE BETWEEN MAN AND MACHINE 1523

the endogenous rate at which ​N​and ​I​grow. The endogenous value of ​n​in an interior
BGP satisfies

​κ​I​​ ι(n) ​v​  CD

​ I​  ​(n)   = ​κN
​ ​​ ​v​  CD
N​  ​(n) .​

The next proposition focuses on interior BGPs and shows that, because of
c­ reative destruction, we must impose additional assumptions on the function ​ι(n)​
to g­ uarantee stability.
_
PROPOSITION 8 (Equilibrium with Creative Destruction): Suppose that ​ρ > ​ρ  ​​,
Assumptions ​1​ ′​, 2, and 4 hold, and there is creative destruction of profits. Then,
_ _ _
There exist ​​ι ​​ and ​​ι_​ < ​ι ​​ such that if ​ι (0) < ​ι_​​ and ​ι (1) > ​ ​​
(i)  ι , then there is
_
at least one locally stable interior BGP with n​ (t ) = n ∈ (​  n ​( ρ), 1).​
_
(ii) If ι​ (n)​ is constant, there is no stable interior BGP (with ​n(t ) = n ∈ (​  n ​( ρ), 1)​).
Any stable BGP involves ​n(t ) → 0​or ​n(t ) → 1​.

The first part of the proposition follows from an analogous argument to that in the
proof of Proposition 6, with the only difference being that, because of the presence
of the function ι​ (n)​in equation (30), the key condition that pins down n​ ​ becomes​​
κ​I​​ ι(n) ​v​  CD
I​  ​(n) = ​κN
​ ​​ ​v​  CD
N​  ​(n)​.
The major difference with our previous analysis is that creative destruction
­introduces a new source of instability. Unlike the previous case with no creative
destruction, we now have that ​​v​  CD I​  ​(n)​is decreasing in ​n​. As more tasks are ­automated,
the rental rate remains unchanged and newly-automated tasks will be replaced less
frequently (recall that newly-automated tasks are replaced after ​(1 − n )/Δ​ units
of time). As a result, automating more tasks renders further a­utomation more
­profitable. Moreover, ​​v​  CD N​  ​(n)​continues to be increasing in ​n​. This is for two ­reasons:
first, as before, the productivity effect ensures that the effective wage in new tasks,​​
w​N​​ (n)​, is decreasing in n​ ​; and second, because newly-created tasks are automated
after ​n/Δ​units of time, an increase in n​ ​increases the present discounted value of
profits from new tasks. These observations imply that, if ​ι(n)​were constant, the
intersection between the curves ​​κN​ ​​v​  CD N​  ​(n)​and ​​κI​​ι(n) ​v​  I​  ​(n)​would correspond to
CD

an unstable BGP.

Economically, the instability is a consequence of the fact that, in contrast to our

baseline model (and the socially planned economy which we describe in the next
subsection), here innovation incentives depend on the total revenue that a ­technology
generates rather than its incremental value created (the difference between these
revenues and the revenues that the replaced technology generated). In our baseline
model, the key force ensuring stability is that incentives to automate are shaped by
the cost difference between producing a task with capital or with labor : by ­lowering
the effective wage at the next tasks to be automated, current automation reduces the
incremental value of additional automation. This force is absent when innovators
destroy the profits of previous technology monopolists because they no longer care
about the cost of production with the technology that they are replacing.
1524 THE AMERICAN ECONOMIC REVIEW JUNE 2018

C. Welfare

We study welfare from two complementary perspectives. First, in online

Appendix B, we discuss the socially optimal allocation in the presence of
­endogenous technology and characterize how this allocation can be decentralized.
One of the main insights from Proposition 6 is that the expected path for factor
prices ­determines the incentives to automate and create new tasks. We show that a
­planner would also ­allocate scientists according to the same principle—guided by
the cost savings that each technology grants to firms. Although similar to the ­efficient
­allocation of s­ cientists in this regard, the decentralized equilibrium is ­typically inef-
ficient because the technology monopolists neither capture the full benefits from the
new tasks they create nor internalize how their innovation affects other existing and
future ­technology monopolists.
The second perspective is more novel and relevant to current debates about
­automation reducing employment and its policy implications. We examine whether
an exogenous increase in automation could reduce welfare. Even though ­automation
expands productivity, a force which always raises welfare, it also reduces ­employment.
When the labor market is fully competitive as in our ­baseline model, this reduction
in employment has no first-order welfare cost for the r­epresentative household (who
sets the marginal cost of labor supply equal to the wage). Consequently, automa-
tion increases overall welfare. Next suppose that there are labor market frictions. In
particular, suppose that there exists an upward-sloping quasi-labor supply schedule,​​
L​qs​​ (ω)​ , which constrains the level of employment, so that ​L ≤ ​Lq​s​​ (ω)​ (see online
Appendix B for a microfoundation). This ­quasi-labor supply schedule then acts in
a very similar fashion to the labor supply curve derived in (11) in Section I, except
that the marginal cost of labor supply is no longer equated to the wage. Crucially, the
reduction in employment resulting from automation now has a negative impact on
welfare, and this negative effect can exceed the positive impact following from the
productivity gains, turning automation, on net, into a negative for welfare.
The next proposition provides the conditions under which automation can reduce
welfare in the context of our static model with exogenous technology. Our focus
on the static model is for transparency. The same forces are present in the dynamic
model and also in the full model with endogenous technology.

PROPOSITION 9 (Welfare Implication of Automation): Consider the static

~
­economy and suppose that Assumptions 1, 2, and 3 hold, and that I​* = I < ​ ​​I .
Let 
​ = u(C, L)​denote the welfare of representative household.

(i) 
Consider the baseline model without labor market frictions, where the
­representative household chooses the amount of labor without constraints
and thus ​W/C = ν ′(L)​. Then,

σ​(( γ(I) ) )
1−​ˆ
σ​
​B​​  ​ˆσ​−1 ​​​ ​ _
​​_ ( )  ​ ​_____ ​  W ​ ​​​  ​− ​R​​  1−​ˆσ​​ ​ > 0,
d ​ = ​ C​e​​  −ν(L)​ ​​​  1−θ
dI 1 − ​ˆ

ˆ( ( γ(N) ) )
ˆ
1−​σ​
​B​​  σ​​ˆ−1 ​​​ ​R​​  1−​σ​
​ _ ( )  ​ ​_____ ˆ​− ​​ _
d ​ = ​​ C​e​​  −ν(L)​ ​​​  1−θ
​  W ​ ​​​  ​ ​ > 0.​
dN 1 − ​σ​
VOL. 108 NO. 6 ACEMOGLU AND RESTREPO: THE RACE BETWEEN MAN AND MACHINE 1525

(ii) 
Suppose that there are labor market frictions, so that employment is
constrained by a quasi-labor supply curve ​L ≤ ​Lq​s​(ω)​. Suppose also
­
that the q­uasi-labor supply schedule ​L ​ ​qs​(ω)​ is increasing in ω
​ ​  , has an
­elasticity ​​~ε​​​  L​​ > 0​  , and is binding in the sense that ​W/C > ν ′ (L)​. Then,

~​​ ε​​ ​​
[ 1 − ​σ​ ˆ (( γ(I) ) ) ​ˆσ​ + ​​ ε​​L​​ ]
​ˆ−1
1−θ ​B​​  σ​ 1−​ˆσ​
​​ _​ = (​ C​e​​  −ν(L)​)​​​  ​​ ____ ​  W ​ ​​​  ​− ​R​​  1−​ˆσ​​ ​− L​(_
​  W ​− ν ′(L))_____
d ​ ​​ ​ _
​  ​​  L~  ​  ​ΛI​ ​​ ​  ≶ 0,
dI C

​​~​​ ε ​​
[ ˆ( ( γ(N) ) ) ​ˆσ​ + ​​ ε​​L​​ ]
1−​ˆσ​
​  W ​ ​​​  ​ ​+ L​(_
​  W ​− ​ν′ ​(L))_____
1−θ ​B​​  ​ˆσ​−1​
​_​ = (
d
​ C​e​​  −ν(L)​)​​​  ​​ ___
​  1 − ​σ​ ​​ ​R​​  1−​ˆσ​​− ​​ _ ​​  L~  ​  ​ΛN​ ​​ ​ > 0.​
dN C
The first part of the proposition shows that both types of technological
i­ mprovements increase welfare when the labor market has no frictions. In this case,
automation increases productivity by substituting cheaper capital for human labor,
and this leads to less work for workers, but since they were previously choosing
labor supply optimally, a small reduction in employment does not have a first-order
impact on welfare, and overall welfare increases. The implications of the creation of
new tasks are similar.
The situation is quite different in the presence of labor market frictions, ­however,
as shown in the second part. Automation again increases productivity and reduces
employment. But now, because workers are constrained in their labor supply choices,
the lower employment that results from automation has a first-order ­negative effect on
their welfare. Consequently, automation can reduce welfare if the productivity gains,
captured by the first term, are not sufficiently large to compensate for the ­second,
negative term. Interestingly, in this case new tasks increase ­welfare even more than
before, because they not only raise productivity but also expand e­ mployment, and
by the same logic, the increase in labor supply has a welfare b­ enefit for the workers
(since they were previously constrained in their employment).
An important implication of this analysis emphasized further in online Appendix B
is that when labor market frictions are present and the direction of t­ echnological change
is endogenized, there will be a force toward excessive a­ utomation. In ­particular, in this
case, assuming that labor market frictions also ­constrain the social planner’s choices,
the decentralized equilibrium involves too much effort being devoted to improving
automation relative to what she would like, because the social planner recognizes that
additional automation has a ­negative effect through employment.

V. Conclusion

As automation, robotics, and AI technologies are advancing rapidly, concerns

that new technologies will render labor redundant have intensified. This paper
develops a comprehensive framework in which these forces can be analyzed and
contrasted. At the center of our model is a task-based framework. Automation is
modeled as the (endogenous) expansion of the set of tasks that can be performed
by capital, r­ eplacing labor in tasks that it previously produced. The main new fea-
ture of our framework is that, in addition to automation, there is another type of
technological change c­ omplementing labor. In our model, this takes the form of the
­introduction of new, more complex versions of existing tasks, and it is assumed that
1526 THE AMERICAN ECONOMIC REVIEW JUNE 2018

labor has a ­comparative advantage in these new tasks. We characterize the structure
of ­equilibrium in such a model, showing how, given factor prices, the allocation of
tasks between capital and labor is determined both by available technology and the
endogenous choices of firms between producing with capital or labor.
One attractive feature of task-based models is that they highlight the link between
factor prices and the range of tasks allocated to factors: when the equilibrium range
of tasks allocated to capital increases (for example, as a result of automation), the
wage relative to the rental rate and the labor share decline, and the equilibrium wage
rate may also fall. Conversely, as the equilibrium range of tasks allocated to labor
increases, the opposite result obtains. In our model, because the supply of labor is
elastic, automation also tends to reduce employment, while the creation of new tasks
increases employment. These results highlight that, while both types of ­technological
changes undergird economic growth, they have very different ­implications for the
factor distribution of income and employment.
Our full model endogenizes the direction of research toward automation and
the creation of new tasks. If in the long run capital is very cheap relative to labor,
automation technologies will advance rapidly and labor will become redundant.
However, when the long-run rental rate of capital is not so low relative to labor, our
framework generates a BGP in which both types of innovation go h­ and-in-hand.
Moreover in this case, under reasonable assumptions, the dynamic equilibrium is
unique and converges to the BGP. Underpinning this stability result is the impact
of relative factor prices on the direction of technological change. The task-based
­framework,  ­differently from the standard models of directed technological change
based on factor-augmenting technologies , implies that as a factor becomes cheaper,
this not only influences the range of tasks allocated to it, but also g­ enerates i­ ncentives
for the introduction of technologies that allow firms to utilize this f­actor more
­intensively. These economic incentives then imply that by reducing the e­ ffective
cost of labor in the least complex tasks, automation discourages further automation
and generates a self-correcting force toward stability.
We show in addition that, though market forces ensure the stability of the BGP,
they do not necessarily generate the efficient composition of technology. If the
­elastic labor supply relationship results from rents (so that there is a wedge between
the wage and the opportunity cost of labor), there is an important new distortion:
because firms make automation decisions according to the wage rate, not the lower
opportunity cost of labor, there is a natural bias toward excessive automation.
Several commentators are further concerned about the inequality implications of
automation and related new technologies. We study this question by extending our
model so that high-skill labor has a comparative advantage in new tasks relative to
low-skill labor. In this case, both automation (which squeezes out tasks previously
performed by low-skill labor) and the creation of new tasks (which directly benefits
high-skill labor) increase inequality. Nevertheless, the long-term implications of the
creation of new tasks could be very different, because they are later standardized and
used by low-skill labor. If this standardization effect is sufficiently powerful, there
exists a BGP in which not only the factor distribution of income (between capital
and labor) but also inequality between the two skill types stays constant.
We consider our paper to be a first step toward a systematic investigation of differ-
ent types of technological changes that impact capital and labor differentially. Several
VOL. 108 NO. 6 ACEMOGLU AND RESTREPO: THE RACE BETWEEN MAN AND MACHINE 1527

areas of research appear fruitful based on this first step. First, our model imposes
that it is always the tasks at the bottom that are automated; in reality, it may be those
in the middle (e.g., Acemoglu and Autor 2001). Incorporating the ­possibility of such
“middling tasks” being automated is an important ­generalization, though ensuring
a pattern of productivity growth consistent with balanced growth in this case is
more challenging. Second, there may be technological barriers to the ­automation
of certain tasks and the creation of new tasks across industries (e.g., Polanyi 1976;
Autor, Levy, and Murnane 2003). An interesting step is to ­construct ­realistic ­models
in which the sectoral composition of tasks performed by capital and labor as well
as technology evolves endogenously and is subject to ­industry-level ­technological
­constraints (e.g., on the feasibility or speed of ­automation). Third, in this paper we
have focused on the creation of new l­ abor-intensive tasks as the type of t­ echnological
change that complements labor and plays a countervailing role against a­ utomation.
Another interesting area is to theoretically and empirically investigate ­different types
of technologies that may complement labor. Fourth, our analysis of the ­creation of
new tasks and standardization abstracted from the need for workers to acquire new
skills to work in such tasks. In practice, the acquisition of new skills may need
to go hand-in-hand with workers shifting to newer tasks, and the i­nability of the
educational system to adapt to the requirements of these new tasks could become
a bottleneck and prevent the rebound in the demand for labor following a wave of
automation. Finally, and perhaps most important, our model highlights the need for
additional empirical evidence on how automation impacts employment and wages
(which we investigate in Acemoglu and Restrepo 2017a) and how the ­incentives
for ­automation and the creation of new tasks respond to policies, factor prices, and
supplies (some aspects of which are studied in Acemoglu and Restrepo 2018b).

Appendix A: Proofs

A. General Model

The analysis in the text was carried out under Assumption 2, which imposed​
η → 0​or ​ ζ = 1​ , and significantly simplified some of the key expressions.
Throughout the Appendix, we relax Assumption 2 and replace it with the following.

ASSUMPTION 2′: One of the following three conditions holds: (i) ​η → 0​; (ii)​
ζ = 1​; or (iii)

max{1, σ}
γ (N − 1)
( γ (N) )
(A1) |σ − ζ| < ​​​ ________
​   ​​​​  ​​ ________________
​​    1
|1 − ζ|
​​.
​​(_____
​  γ(N − 1) ​)​​​ 
γ (N)
​ −1

All of our qualitative results remain true and will be proved under this more ­general
assumption. Intuitively, the conditions in Assumption 2 ensured ­ homotheticity
(see footnote 12). Assumption ​2′​, on the other hand, requires that the departure from
homotheticity is small relative to the inverse of the productivity gains from new
tasks (where γ ​ (N )/γ (N − 1)​measures these productivity gains).
1528 THE AMERICAN ECONOMIC REVIEW JUNE 2018

Task prices in this more general case are given by

(A2) ​p(i ) =

⎧
⎪​c​​  ​​(min​{R, _
_
​  1  ​

{ γ(i) } ]
γ(i) }) [
1−ζ 1−ζ
​+ (1 − η) min ​​ R, _
u
​  W ​ ​ ​ = ​​ η ​ψ​​ 
​  W ​ ​​​  ​ ​​​  ​ if i ≤ I 1−ζ

⎨​       
⎪​c​​  ​​ _
​ ​  ​  ​.​​​
1 _
​   ​

⎩ ( γ(i) ) [ ( γ(i) ) ]
1−ζ 1−ζ
u
​  W ​ ​ = ​​ η ​ψ​​  1−ζ
​+ (1 − η) ​​ _
​  W ​ ​​​  ​ ​​​  ​ if i > I

Here ​​c​​  u​( ⋅ )​is the unit cost of production for task ​i​ , derived from the task ­production
functions, (2) and (3). Naturally, this equation simplifies to (5) under Assumption 2.
From equations (5) and (7), equilibrium levels of task production are

⎧ ​ˆσ​−1 u −σ

⎪ ( { γ(i) })
​
B ​​  ​Y​c ​​  ​​​ min​ R, _
​  W ​ ​ ​​​  ​ if i ≤ I
​ ⎨​    
y(i ) = ​ ​  ​  ​​​ ​.
⎪​B​​  σ​
−σ

( γ(i) )
​ˆ−1
​Y​c​​  u​​​ _ ​  W ​ ​​​  ​ if i > I
⎩
Combining this with equations (2) and (3), we obtain the task-level demands for
capital and labor as

{0
​B​​  ​ˆσ​−1​  (1 − η ) Y​c​​  u​​(R)​​  ζ−σ​​R​​  −ζ​ if i ≤ I *
​ k(i ) = ​ ​    
​ ​  ​​ ​​
if i > I *
and

⎧0 if i ≤ I *
⎪
⎨​     
l(i ) = ​ ​ ζ−σ ​   ​​​ ​.
( γ(i) )
​
⎪​B​​  ​ˆσ​−1​(1 − η ) Yγ ​(i)​​  ζ−1​​c​​  u​​​ _
​  W ​ ​​​  ​​W​​  −ζ​ if i > I*
⎩

Aggregating the preceding two equations across tasks, we obtain the following
capital and labor market-clearing equations,

(A3) ​​B​​  σ​​ˆ−1​  (1 − η ) Y( ​I​​  ∗​− N + 1) ​c​​  u​​(R)​​  ζ−σ​​R​​  −ζ​ = K, ​

and

(A4) ​​B​​  σ​​ˆ−1​  (1 − η ) Y ​∫ *​  ​​ γ ​(i )​​  ζ−1​​c​​  u​​​ _

ζ−σ

( γ(i) )
​  W ​ ​​​  ​ ​​  s​​(_
​​W​​  −ζ​di = L ​  W )
​ ​.​
N
I RK
Finally, from the choice of aggregate output as the numéraire, we obtain a
­generalized version of the ideal price condition,

( I* − N + 1) ​c​​  u​​(R)​​  1−σ​+ ​∫ *​  ​​ ​c​​  u​​​ _

( γ(i) )
1−σ
(A5) ​ ​  W ​ ​​​  ​di = ​B​​  1−​ˆσ​​, ​
N
I

which again simplifies to the ideal price index condition in the text, (10), under
Assumption 2.
VOL. 108 NO. 6 ACEMOGLU AND RESTREPO: THE RACE BETWEEN MAN AND MACHINE 1529

B. Proofs from Section II

PROOF OF PROPOSITION 1:
We prove Proposition 1 under the more general Assumption ​​2′​.
To prove the existence and uniqueness of the equilibrium, we proceed in three
steps. First, we show that ​I *​, ​N​, and ​K​  determine unique equilibrium values for​
R, W​, and Y​ ​ , thus allowing us to define the function ω ​ ( I *, N, K)​representing the
relative demand for labor, which was introduced in the text. Second, we prove a
lemma which ensures that ​ω( I *, N, K)​is decreasing in I​*​ (and increasing in N ​ ​).
Third, we show that ​min   ​{I, ​ ​I }​​is nondecreasing in ω
~
​ ​and conclude that there is
​​ ω *, I *}​​such that ​I * = min   ​{I, ​ I​}​​ and ​ω * = ω​( I *, N, K)​​. This pair
~
a unique pair {
uniquely d­ etermines the equilibrium relative factor prices and the range of tasks that
get effectively automated.

Step 1: Consider I​ *, N​, and ​K​such that ​I * ∈ (N − 1, N)​. Then, ​R, W​, and ​Y​ satisfy
the system of equations given by capital and labor market-clearing, equations (A3)
and (A4), and the ideal price index, equation (A5).
Taking the ratio of (A3) and (A4), we obtain

​∫ *​  ​​γ ​(i )​​  ζ−1​​c​​  u​​​ _

ζ−σ

( γ(i ) )
​  W ​ ​​​  ​​W​​  −ζ​di
N
I
____________________________
   ​ =
(A6) ​​    _
​  1 ​.​
​L​​  s​​(_​  W )
K
​ ​(I * − N + 1) ​c​​  u​​(R)​​  ζ−σ​​R​​  −ζ​
RK
In view of the fact that ​L ​ ​​  s​​is increasing and the function ​c​ ​​  u​​(x)​​  ζ−σ​​x​​  −ζ​​is decreasing in​
x​ (as it can be verified directly by differentiation), it follows that the left-hand side
is decreasing in ​W​and increasing in ​R​. Therefore, (A6) defines an u­ pward-sloping
relationship between W ​ ​and ​R​, which we refer to as the relative demand curve
(because it traces the combinations of wage and rental rate consistent with the
demand for labor relative to capital being equal to the supply of labor divided by
the capital stock).
On the other hand, inspection of equation (A5) readily shows that this equation
gives a downward-sloping locus between ​R​and ​W​as shown in Figure A1, which we
refer to as the ideal price curve.
The unique intersection of the relative demand and ideal price curves determines
the equilibrium factor prices for given ​I *​, ​N​, and ​K​. Because the relative demand
curve is upward-sloping and the ideal price index curve is downward-sloping, there
can be at most one intersection. To prove that there always exists an i­ntersection,
observe that ​ ​lim​x→0​​ ​c​​  u​​(x)​​  ζ−σ​​x​​  −ζ​ = ∞​, and that ​ ​lim​x→∞​​ ​c​​  u​​(x)​​  ζ−σ​​x​​  −ζ​ = 0​.
These ­observations imply that as W ​ → 0​, the numerator of (A6) limits to infinity,
and so must the denominator, i.e., ​R → 0​. This proves that the relative demand
curve starts from the origin. Similarly, as W ​ → ∞​, the numerator of (A6) ­limits
to zero, and so must the denominator (i.e., ​R → ∞​). This then implies that the
­relative demand curve goes to infinity as R ​ → ∞​. Thus, the upward-sloping ­relative
demand curve ­necessarily starts below and ends above the ideal price curve, which
ensures that there always exists an intersection between these curves. The unique
1530 THE AMERICAN ECONOMIC REVIEW JUNE 2018

W Relative labor demand

W (I, N, K )

Ideal price index

R (I, N, K ) R

Figure A1. Construction of the Function ​ω( ​I​​ *, N, K )​

intersection defines the equilibrium values of W

​ ​and R
​ ​, and therefore the function​
* _
ω( I , N, K ) = ​ ​​  .
W
RK
Step 2: This step follows directly from the following lemma, which we prove in
online Appendix B.
_ ~
LEMMA A1: Suppose that Assumption ​ 2′​ holds, ​K < ​  K ​​ and ​I * ≤ ​ I​​.
Then ​ω( I *, N, K )​is decreasing in ​I *​and is increasing in N
​ ​.

Although the general proof for this lemma is long (and thus relegated to online
Appendix B), the lemma is trivial under Assumption 2. In that case, equation (A6)
yields
∫
​ ​  ​​γ ​(i)​​  ​ˆσ​−1​  di
N

ω ​( I *, N, K)​​  ​ˆσ​​​L​​  s​(ω( I *, N, K )) ___________

= ​     ​K​​  1−​σ​ˆ​.​
 ​
I *
​
I* − N + 1
Taking logs, we obtain equation (13) in the main text, which implies that ω
​ ( I *, N, K )​
*
​ ​and decreasing in I​ ​.
is increasing in N

Step 3: We now show that I​* = min  ​{I, ​ I​}​​is uniquely defined. Because

~
​γ ​(​ I​)​ = ωK​, we have that I​* = min  ​{I, ​ I​  }​​is increasing in ω
~ ~
​ ​and has a vertical
asymptote at I​ ​.
​ = ω​(I *, N, K )​​ and ​I * = min  ​{I, ​ I​}​​ plot-
~
Consider the pair of equations ω
​ = ω​(I *, N, K )​​is decreasing in I​*​for ​I * ≤ ​ ​​
~
ted in Figure 3. Because ω I and ​I *
= min ​{I, ​ I​}​is increasing in ω
​ ​ , there exists at most a unique (​​ ω, I )​​ satisfying these
~ *

two equations (or a unique intersection in the figure).

To prove existence, we again verify the appropriate boundary conditions. Suppose
that I​* → N − 1​. Then from (A3), ​R → 0​ , while ​W > 0​ , and thus ​ω → ∞​.
This ensures that the curve ​ω ​(I *, N, K)​​starts above I​* = min ​{ I, ​ I​}​​ in Figure 3.
~
Since I​* = min  ​{I, ​ ​I}​​has a vertical asymptote at ​I < N​, the two curves must
~
­intersect. This observation completes the proof of the existence and uniqueness of
the equilibrium.
VOL. 108 NO. 6 ACEMOGLU AND RESTREPO: THE RACE BETWEEN MAN AND MACHINE 1531

When Assumption 2 holds we can explicitly solve for aggregate output. In this
case, the market-clearing conditions, (8) and (9), become

__
​  1  ​
− η )​∫ *​ 
__
Y  ​ ​​​  ​  σ ​,​
1
W =  ​​(
L)
​ˆ
​
σ
​R = ​​( ​B​​  ​ˆσ​−1​  (1 − η) (I* − N + 1) ​ __ ) ​B​​  σ​ˆ​ −1​  (1 ˆ
​ −1 _
​​ γ ​(i )​​  ​  di ​  ​ ​​​  ​,​
σ​
N Y
K I

which combined with (10) yields (12), completing the proof of Proposition 1. ​∎​

C. Proofs from Section III

LEMMA A2 (Derivation of Figure 4): Suppose that Assumptions ​​1′​and ​​2′​ hold.
Consider a path of technology where ​n(t ) → n​ and ​g(t ) → g​, consumption grows
_
at the rate ​g​and the Euler equation (17) holds. Then, there exist ​ρ
​m​ in​​ < ​ρ
 ​ < ​ρ​max​​​
such that:
_ _
(i) If ​ρ ∈ [ ​ρ​min​​ , ​ρ
 ​]​  , there is a decreasing function ~ ​​ n ) : [ ​ρm
​(ρ ​ in​​ , ​ρ
 ​ ]   → (0, 1]​
such that for all ​n > ​ ~ n ρ)​ we have ​w
​( ​ I​​​ (n ) > ρ + δ + θg > ​w​N​​ (n)​ and​
_
ρ + δ + θg = ​wN​ ​​ (​ ~ n​( ρ ))​. Moreover, ~
n​​  ​ρm
​( ​ in​​ ) = 1​ and ~​​ n ρ
​(​  ​) = 0​.
_ _ _
(ii) If ​ρ ∈ [ ​ρ
 ​, ​ρm
​ ax​​ ]​, there is an increasing function ​​  n ​( ρ ) : [ ​ρ  ​, ​ρm​ ax​​ ]   → (0, 1]​
_
such that for all n​ > ​  n ​(ρ)​  , we have ​w ​ I​​​ ( n ) > ρ + δ + θg > ​wN​ ​​ (n)​ and​
_ _ __
ρ + δ + θg = ​w​I​​ (​  n ​( ρ ))​. Moreover, ​​  n ​( ​ρm​ ax​​ ) = 1​and ​​  n ​(​ρ
 ​) = 0​.

​ ax​​​  , for all n​ ∈ [0, 1]​ we have ρ

(iii) If ​ρ > ​ρm ​ + δ + θg > ​wI​​​ ( n ) ≥ ​wN​ ​​ ( n )​,
which implies that automation is not profitable for any ​n ∈ [0, 1]​.

​ in​​​  , for all n​ ∈ [0, 1]​ we have ​​w​I​​ (n ) ≥ ​wN​ ​​ (n ) > ρ + δ + θg​,
(iv) If ​ρ < ​ρm
which implies that new tasks do not increase aggregate output and will not be
adopted for any ​n ∈ [0, 1]​.

PROOF:
Because consumption grows at the rate ​g​and the Euler equation (17) holds,
we have

R(t ) = ρ + δ + θg.​
​

The effective wages ​​wI​​​ (n)​and ​w

​ N​ ​​ (n)​are then determined by the generalized ideal
price index condition, (A5), as

∫0 ​w​​​ (n)
( γ(i ) )
n 1−σ
(A7) ​​B​​  1−​σˆ​​ = (1 − n) ​c​​  ​​( ρ + δ + θg)​​ 
u 1−σ
​+ ​ ​  ​​c​​  ​​ _
u
​  I ​ ​​​  ​di

= (1 − n) ​c​​  u​​(ρ + δ + θg)​​  1−σ​+ ​∫ ​  ​​c​​  u​​(γ(i ) ​wN​ ​​ ( n ))​​  1−σ​di.​

n

0
1532 THE AMERICAN ECONOMIC REVIEW JUNE 2018

Differentiating these expressions, we obtain

​w​  ′I​(​  n) 1  ​​(​c​​  u​​(ρ + δ + θg)​​  1−σ​− ​c​​  u​​(​w​ ​​(n))​​  1−σ​)​

(A8) ​​____ ​ = ​ ____
​wI​​​(n) 1−σ N

× __________________________________
​      1
∫
   n ​  ,
​ 0​  ​​ ​c​​  ​( ​wN​ ​​ (n ) γ(i )) ​c​​  ​​(​wN​ ​​ (n) γ (i ))​​  ​​wN​ ​​ (n) γ (i ) di
u′ u −σ

​w​  ′N​(​  n)
_____
​ ​ 1 (​​ ​c​​  u​​(ρ + δ + θg)​​  1−σ​− ​c​​  u​​( ​wI​​​ (n ))​​  1−σ​)​
​= _
​wN​ ​​(n) 1−σ

× ​____________________________________
      1
∫
     ​.​
u′ ​
n
​ 0​  ​​ ​​c​​  ​​​  ​( ​wN​ ​​ (n ) γ (i )) ​c​​  ​​(​wN​ ​​ (n) γ(i ))​​  −σ​​wN​ ​​ (n ) γ (i ) di
u

To prove part (i), define ​ρ ​ ​​ as

​ min

​ρ​min​​ + δ + θg = ​wN​ ​​ (1),​

​
_
and define ​​ρ
 ​ > ​ρ​min​​​ as
_ ˆ​.​
​c​​  u​​(​ρ
​  ​ + δ + θg)​​  1−σ​ = ​B​​  1−​σ​
_
(When Assumption 2 holds, we get ​​ρ  ​ = B − δ − θg​, as claimed in the main text.)
_
To show that ​​ρ
 ​ > ​ρm
​ in​​​ , note that

​​_ ​ 1 ​∫
1 ​​c​​  u​​( ​ρ​ ​​ + δ + θg)​​  1−σ​ = _ ​ 0​  ​​c​​  u​​( ​ρ​min​​ + δ + θg)​​  1−σ​di
1
1−σ min
1−σ

​ 1 ​∫
= _
1
​ ​  ​​c​​  u​​( ​wN​ ​​ (1))​​  1−σ​di
1−σ 0

​ 1 ​∫
< _
1
​ ​  ​​c​​  u​​(​wN​ ​​ (1) γ (i))​​  1−σ​di
1−σ 0

= ​_1 ​B ˆ​
​ ​​  1−​σ​
1−σ
_
= _
​ 1 ​​c​​  u​​(​ρ
 ​ + δ + θg)​​  1−σ​.​
1−σ
_
Because the function _
​​ 1 ​ ​c​​  u​​(x)​​  1−σ​​is increasing, we have ​​ρ​ > ​ρ​min​​.
1−σ

​​  (ρ)​ implicitly

Using the generalized ideal price index condition, (A5), we define ~
n​
as

​c​​  u​​( ρ + δ + θg)​​  1−σ​+ ​∫0​  ​​ ​c​​  u​​(γ (i ) (ρ + δ + θg))​​  1−σ​di.​

ˆ​ = (1 − ​ ~ n (ρ) ~
​​B​​  1−​σ​
 ​ ​
​(ρ))
n
VOL. 108 NO. 6 ACEMOGLU AND RESTREPO: THE RACE BETWEEN MAN AND MACHINE 1533

Differentiating this expression with respect to ρ ​ ​shows that ~ ​​n​(ρ)​is decreasing.

_ _
Moreover, ​​n​( ​ρm ~ ​ in​) = 1​ and ​​n​(​ρ​) = 0​  , so n​
~ ​​ ( ⋅ )​is well defined for ​ρ ∈ [ ​ρm
~ ​ in​, ​ρ​]​.
For ​n = ​ ~n​(ρ)​, we have ​​wI​​(​~
n​(ρ )) > ρ + δ + θg = ​wN​ ​(​~ n​( ρ))​. Thus, the ­formulas
_
for ​​w​  ′I​ ​(n)​ and ​​w​  ′N​​  (N)​show that, for ​ρ ∈ [ ​ρm ​ in​, ​ρ​]​and starting at ~ ​​n​(ρ)​  , the curve​​
w​N​​ (n)​is decreasing in ​n​and the curve ​​w​I​(n)​is increasing in ​n​. Thus, for all ​n > ​ n​(ρ)​  ,
~
we have

​wI​​​ (n) > ​wI​​​ (​ ~

​ n​( ρ )) > ρ + δ + θg = ​wN​ ​​ (​ ~
n​( ρ ))   > ​wN​ ​​ (n),​

as claimed. On the other hand, for all n​ < ​ ~ n ρ )​, we have ​​wN​ ​​ (n) > ρ + δ + θg​.
​(
_
​ ax​​ > ​ρ
To prove part (ii), define ​​ρm  ​​ as

​ρm
​ ​ ax​​ + δ + θg = ​wI​​​ (1).​
_
To show that ​​ρ
 ​ < ​ρm
​ ax​​​ , a similar argument establishes

​​_ ​ 1 ​∫
1 ​​c​​  u​​( ​ρ​ ​​ + δ + θg)​​  1−σ​ = _ 1
​ ​​ + δ + θg)​​  1−σ​di
​ ​  ​​c​​  u​​( ​ρmax
1−σ max
1−σ 0

​ 1 ​∫
> _
1
​ ​  ​​c​​  u​​(​wI​​​ (1)/ γ (i))​​  1−σ​di
1−σ 0
_
= _
​ 1 ​​c​​  u​​( ​ρ
 ​ + δ + θg)​​  1−σ​.​
1−σ
_
Because the function _
​​ 1 ​ ​c​​  u​​(x)​​  1−σ​​is increasing, we have ​​ρ​ < ​ρm
​ ax​​.
1−σ _
Using (A7), we define the function ​​n ​(ρ)​implicitly as
_
​​B​​  1−​ˆσ​​ = (1 − ​n ​( ρ)) ​c​​  u​​( ρ + δ + θg)​​  1−σ​+ ∫
_ ​  n ​( ρ)
​ 0​  ​​c​​  u​​(( ρ + δ + θg)/γ( i ))​​  1−σ​di.​
_
Differentiating this expression with respect to ​ρ​ shows that_​​  n ​(ρ)​is increasing in ​ρ​
_ _ __
on ​[  ​ρ
 ​, ​ρm
​ ax​​ ]​. Moreover, ​​  n ​( ​ρ​max​​ ) = 1​and ​​  n ​(​ρ  ​) = 0​  , so ​​  n ​( ⋅ )​is well defined for
_
all ρ ​ ≥ ​ρ  ​​  .
_
For ​ n = ​  n ​(ρ)​ , we have ​ ​wI​​​ (​ ~
n​( ρ )) = ρ + δ + θg > ​wN​ ​​ (​ ~ n​( ρ ))​. Thus, the
_ _
­formulas for ​​w​  ′I​(​  n)​ and ​​w​  N′ ​​  (n)​show that, for ρ ​ ∈ [ ​ρ
 ​, ​ρmax
​ ​​ ]​and starting at ​ ​n ​(ρ)​,
_
​​w​N​​ (n)​is decreasing in n​ ​and ​w ​ ​I​​ (n)​is increasing in n​ ​. Thus, for all n​ > ​  n ​( ρ )​,
we have
_ _
​wI​​​ (n) > ​wI​​​ (​  n ​( ρ ))   = ρ + δ + θg > ​wN​ ​​ (​  n ​( ρ ))   > ​wN​ ​​ (n ),​
​
_
as claimed. On the other hand, for all ​n < ​  n ​( ρ )​ , we have ​​wI​​​ (n ) < ρ + δ + θg​.
_
In this region we have ​n * = ​  n ​( ρ) > n​ , and not all automated tasks are produced
with capital.
To prove part (iii), note that for ​ρ > ​ρm ​ ax​​​ , we have

ρ + δ + θg > ​w​I​​ (1) > ​wN​ ​​ (1).​

​
1534 THE AMERICAN ECONOMIC REVIEW JUNE 2018

The expressions for ​​w​  ′I​  (n)​​ and ​​w​  ′N​(​  n)​show that in this region, as ​n​decreases, so
does ​​wI​​​ (n)​. Thus, ​ρ + δ + θg > ​w​I​​ ( n ) > ​wN​ ​​ (n)​, and for all these values we have​
n* = 1​ , and no task will be produced with capital.
To prove part (iv), note that for ​ρ < ​ρm ​ in​​​, we have

ρ + δ + θg < ​w​N​​ (1) < ​wI​​​ (1).​

​

The expressions for ​​w​  ′I​​  (n)​ and ​​w​  ′N​​  (N)​show that, in this region, as ​n​ decreases, both​​
w​I​​ (n)​, ​​w​N​​ (n)​increase. Thus, ρ
​ + δ + θg < ​w​N​​ (n) < ​wI​​​ (n)​and for these v­ alues of​
ρ​ , new tasks do not raise aggregate output. ∎

PROOF OF PROPOSITION 4:
We prove this proposition under the more general Assumption ​​2′​.
We start by deriving necessary conditions on ​N(t)​and ​I(t)​such that the economy
admits a BGP, and then show that these are also sufficient for establishing the exis-
tence of a unique and globally stable BGP.
The capital market-clearing condition implies that
K(t)
​c​​  u​​(R(t))​​  σ−ζ​R ​(t)​​  ζ​_
​ ​  ​ = ​B​​  σ​​ˆ−1​  (1 − η ) (1 − n*(t)).​
Y(t)
Because in BGP the rental rate of capital, R ​ (t)​
, and the capital to aggregate
​ (t)/Y(t)​ , are constant, we must have n​ *(t ) = n​, or in other words,
output ratio, K
labor and capital must perform constant shares of tasks.
Lemma A2 shows that we have four possibilities corresponding to the four cases
in Proposition 4, each of which we now discuss in turn.

(i) All Tasks Are Automated: n​ *(t) = n = 0​. Because in this case c­apital­
_
performs all tasks, Lemma A2 implies that we must have ρ ​ < ​ρ ​​ and
​ ​K​​ K​  ,
​I(t ) = N(t)​. In this part of the parameter space, net output is given by ​A
​A​ ​​ − δ − ρ
and the e­ conomy grows at the rate ______
​​ K θ ​​. The transversality condition, (19),
​A​ ​​ − δ − ρ
is satisfied if and only if ​ ​AK​ ​− δ > ​______
K
θ
​​ ­
— or ​r > g​. Moreover,
­positive growth imposes ​​A​K​ > δ + ρ​. The generalized ideal price index
ˆ
condition, equation (A5), then implies that ​R = ​c​​  u​​​  −1​ ​( ​B ​​  ​  1−σ ​​)​​, and thus
1−​σ​
____

1−​ˆσ​
​ ​K​ = ​c​​  u​​​  −1​ ​( ​B ​​  ​  1−σ ​​)​​. Under Assumption 2, this last expression further
____
​A
­simplifies to ​​A​K​ = B​as claimed in the text.

We now show that these necessary conditions are sufficient to generate balanced
_
growth. Suppose ​ρ < ​ρ  ​​and I​ (t) = N(t)​so that n​ * (t )   = 0​. Because all tasks are
produced with capital, we also have ​​FL​ ​​ = 0​ , and thus the representative household
supplies zero labor. Consequently, the dynamic equilibrium can be characterized as
the solution to the system of differential equations
​C˙​(t)
​​ ___ ​ = _ ​  1 ​ (​AK​ ​​ − δ − ρ),
C(t) θ
_ θ−1
​ K˙​(t) = ( ​A​K​​ − δ ) K(t) − C(t)​e​​  ν(0)​  θ ​​,​
VOL. 108 NO. 6 ACEMOGLU AND RESTREPO: THE RACE BETWEEN MAN AND MACHINE 1535

together with the initial condition, ​K(0) > 0​and the transversality condition,
(19). We next show that there is a unique solution to the system, and this solution
­converges to the full automation BGP described in the proposition.
​​​ c = C/K​. The behavior of ~
Define ~ c​​  is governed by the differential equation,
​​

​​c˙ ̃​(​  t) θ−1

​​ ___ ​ = _
_
​1 ​(​AK​ ​− δ − ρ) − (​AK​ ​− δ ) + ​ ~c​(t)​e​​  ν(0)​  θ ​​.​
​c̃​(t) θ

This differential equation has a stable rest point at zero and an unstable rest point
at ​c​ ​B​ = ​(​Ak​​​ − δ − _
​  1 ​ ( ​AK​ ​​ − δ − ρ))​​e​​  ν(0)​  θ ​​ > 0​. There are therefore three possible
_ 1−θ
θ
equilibrium paths for ~​​c​(t)​: (i) it immediately jumps to ​​cB​ ​​and stays there; (ii) it starts
at ​[ 0, ​c​B​)​and converges to zero; (iii) it starts at ​(c​ B​ ​, ∞)​and diverges. The ­second and
third possibilities violate, respectively, the transversality condition (19) (because
the capital stock would grow at the rate ​A ​ ​k​​ − δ​ , implying ​r = g​), and the resource
constraint (when ~​​c​(t ) = ∞​). The first possibility, on the other hand, satisfies the
transversality condition (since asymptotically it involves r​ > g​), and yields an
equilibrium path. In this path the economy converges to a unique BGP in which​
​AK​ ​​ − δ − ρ
​ (t )​grow at a constant rate  ​​______
C(t )​and K θ
​​ , thus also establishing uniqueness
and global stability.

(ii) Interior Equilibrium in which Automated Tasks Are Immediately

Produced with Capital: ​
­ n*(t) = n(t) = n ∈ (0, 1)​. Because capital
_
­performs all ­automated tasks, Lemma A2 implies that ​n > max  {​n ​( ρ), ​~ n​( ρ)}​
and ​​N˙ ​(t) = ​I˙​(t)​. Moreover, because in this candidate BGP ​R(t)​ is constant,
the general form of the generalized ideal price index condition, (A5), implies
that ​W(t)/γ (I(t))​must be constant too, and this is only possible if ​​I˙​(t) = Δ​.
Consequently, the growth rate of aggregate output is ​ AΔ​. Finally,
the t­ ransversality condition, (19), is satisfied given the condition​
ρ + (θ − 1 ) AΔ > 0​in this part of the proposition. Lemma A2 then v­ erifies
_
that n​*(t) = n > ​n ​(ρ)​. Substituting the market-clearing ­ conditions
for ­capital and labor, (A3) and (A4), into (1), (2), and (3) and then
­subtracting the costs of intermediates, we obtain net output as ​F(k, L; n)​.
(When Assumption 2 holds, F ​ (k, L; n)​ is given by the CES aggregate in
equation (16)). ​F(k, L; n)​exhibits constant returns to scale, and because
factor markets are c­ ompetitive, we also have R ​ (t) = ​FK​ ​​ (k(t), L(t); n)​ and
​w(t) = ​FL​ ​​ (k (t), L(t); n)​.

To establish uniqueness, let ​​wB​ ​​​denote the BGP value of the wage rate, ​​k​B​​​the BGP
value of the normalized capital stock, ​c​ ​B​​​the BGP value of normalized ­consumption,​​
L​B​​​the BGP value of employment, and ​​RB​ ​​​the BGP value of the rental rate of ­capital.
These variables are, by definition, all constant. Then, the Euler e­ quation, (17), implies​​
R​B​​ = ρ + δ + θg​, and because ​R ​ ​B​​ = ​FK​ ​​ ( ​kB​ ​​, ​LB​ ​​; n)​, we must also have ​k​ B​ ​​/​L​B​​  = ϕ​,
where ϕ is the unique solution to

​F​K​(ϕ, 1 ; n ) = ρ + δ + θg.​
​
1536 THE AMERICAN ECONOMIC REVIEW JUNE 2018

_
Lemma B1 in online Appendix B shows that, for n​ > max  {​  n ​(ρ), ​ñ​(ρ)}​  , ​F(ϕ, 1; n)​
satisfies the following Inada conditions,

​​ lim​​​ ​FK​ ​​ (ϕ, 1; n ) > ρ + δ + θg, ​  lim​​​ ​FK​ ​​ (ϕ, 1; n) < ρ + δ + θg,​
ϕ→0 ϕ→∞

which ensure that ϕ is well defined. Combining the labor supply condition, (18),
_____ ​F​ ​​ (ϕ, 1)
with the resource constraint, (20), we obtain (​ F(ϕ, 1; n ) − (δ + g )ϕ) ​L​B​ = ​ L
ν ′ ( ​L​ ​​)
​​.
B
The left-hand side of this equation is linear and increasing in ​L​ (the concavity of​
F​in k​ ​implies that ​F(ϕ, 1; n) > ϕ ​F​K​(ϕ, 1; n) > (δ + g )ϕ​), while the right-hand
side is decreasing in L ​ B​ ​​ > 0​ that
​ ​ . This ensures that there exists a unique value ​L
­satisfies this equation, and also pins down the value of the normalized capital stock
as ​k​​B​​ = ϕ ​LB​ ​​​ . Finally, ​​cB​ ​​​is uniquely determined from the resource constraint,
(20), as

_
​c​B​ = (F(ϕ, 1; n) − (δ + g )ϕ) ​L​B​​e​​  ν(​LB​ ​​)​   ​
​.​
1−θ
​ θ

Note also that there cannot be any BGP with ​ ​LB​ ​​ = 0​ 
, since this would
imply ​c​​B​​ = 0​from the resource constraint, (20). But then we would have
θ−1 ​F​ ​​ (ϕ, 1; n)
ν​ ′ (0) ​e​​  ​  θ ​ν(0)​ < ​______
___ L
​cB​ ​​ ​​ , which contradicts the labor supply optimality c­ ondition,
(18). Hence, the only possible BGP is one in which k​ (t) = ​kB​ ​​  , ​c(t) = ​cB​ ​​, and​
L(t) = ​L​B​ > 0​ . Moreover, in view of the fact that ρ ​ + (θ − 1 ) AΔ > 0​  , this
­candidate BGP satisfies the transversality condition (19), and is indeed the unique
BGP. The proof of the global stability of this unique BGP is similar to the analysis
of global stability of the neoclassical growth model with endogenous labor supply,
and for completeness, we provide the details in online Appendix B.

(iii) Interior Equilibrium in which Automated Tasks Are Eventually but Not
_
Immediately Produced with Capital: ​ n *(t) = ​  n ​( ρ) > n(t)​. Because
­capital does not immediately perform all automated tasks, Lemma A2 implies
_ _
that n​ (t) < ​  n ​( ρ)​and ​ρ > ​ρ
 ​​. Moreover, because ​R(t)​ is constant, the ideal
price index condition, (A5), implies that ​W(t)/γ( I *(t))​must be constant too.
Thus, to generate constant growth of wages we must have I​​˙​* (t) = Δ ≤ I(t)​,
_
so that the growth rate of the economy is given by ​AΔ​. Because ​n * (t) = ​  n ​( ρ)​,
this also implies that ​​N˙ ​(t) = Δ​  . Finally, the transversality condition, (19),
is satisfied in view of the fact that this part of the proposition imposes​
ρ + (θ − 1) AΔ > 0​. The uniqueness and global stability of the BGP follow
_
from an identical arguments to part (ii), with the only modification that ​​  n ​( ρ)​
plays the role of n​ ​in the preceding proof.

(iv) All Tasks Are Always Produced with Labor: ​​n​​  ∗​(t ) = 1​. Because labor
­performs all tasks, Lemma A2 now implies ​ρ > ​ρ​max​​​ and ​n(t) ≥ 1​, while
the ideal price index condition, (A5), imposes that ​W(t)/γ (N(t))​must be
­constant. Thus, to generate a constant wage, aggregate output and ­capital
growth, we must have ​​N˙ ​(t) = Δ​, with ​ρ + (θ − 1)Δ > 0​ (where the last
­condition again ensures transversality). To show sufficiency of these ­conditions
VOL. 108 NO. 6 ACEMOGLU AND RESTREPO: THE RACE BETWEEN MAN AND MACHINE 1537

for balanced growth, let ​​wB​ ​​​denote the BGP value of the ­normalized wage,
which is defined by

∫ ˆ​.​
​ 0​  ​​ ​c​​  u​​(​wB​ ​​/γ (i))​​  1−σ​ = ​B​​  1−​σ​
​
1

Consequently, net output is given by ​F(k, L; n ) = ​wB​ ​​ γ (N(t) − 1) L(t)​, and

thus depends linearly on labor and is independent of capital. This implies​
K(t) = 0​and ​C(t) = ​wB​ ​​ γ (N(t) − 1) L(t)​. The representative household’s
labor supply condition, (18), implies that in this BGP
​w​ ​​ γ(N(t) − 1)
ν ′(L(t)) = ___________
​ ​  B
   ​ = ​ _1 ​,​
C(t) L(t)
which uniquely defines a BGP employment level ​L ​​B​​​  . Because this
a­ llocation also satisfies the transversality condition (in view of the fact that
​ρ + (θ − 1)AΔ > 0​ ), it defines a unique BGP. Its global stability ­follows by
noting that starting with any positive capital stock, ​K(0) > 0​  the ­representative
­household chooses zero ­investment and converges to this path. ​∎​

D. Proofs from Section IV

All of the results in this section apply and will be proven under Assumption ​​2′​.

LEMMA A3 (Asymptotic Behavior of the Normalized Value Functions): Suppose

​κI​ ​​ ​κN​ ​​
that Assumptions ​​1′​, ​2​ ′​, and 4 hold. Let g​ = A ​_____
​κI​ ​​ + ​κN​ ​​ ​  S​denote the growth rate of the
~ ~
economy in a BGP. Then there exists a threshold S​​ ​​ such that for S​ < ​ S​​, we have​
ρ + (θ − 1)g > 0​, and:

_
• If ​n ≥ max  {​ ​  n ​, ​ ~
n​}​​, both ​​vN​ ​(n)​ and ​​vI​​(n)​ are positive and increasing in n​ ​;
_ _
• If n​ ≤ ​  n ​( ρ)​ (and ​ρ > ​ρ  ​​  ), we have ​​κ​N​​ ​vN​ ​​ (n) > ​κI​ ​​ ​vI​​​ (n) = (g)​ (meaning
that it goes to 0 as ​g → 0​);
_
• If ​n < ​ ~
n​(ρ)​ (and ​ρ < ​ρ  ​​  ), we have ​​κI​ ​​ ​vI​​​ (n) > 0 > ​κN​ ​​ ​vN​ ​​ (n).​ Moreover, in
this region, ​​v​I​​ (n)​is decreasing and ​​vN​ ​​ (n)​is increasing in ​n​  .

PROOF:
See online Appendix B.

PROOF OF PROPOSITION 6:
We first show that all the scenarios described in the proposition are BGPs with
endogenous technology. We then turn to analyzing the stability of interior BGPs.

Part 1: Characterization of the BGPs with Endogenous Technology: Suppose

~
that S​ < ​ ​​
S so that Lemma A3 applies. We consider the two cases described in the
proposition separately.
_
​ < ​ρ
(i) ρ  ​​ : Suppose that ​n < ​ ~
n ρ)​. As depicted in panel A of Figure 8 and
​(
shown in Lemma A3, in this region ​​vI​​​ (n)​is positive and decreasing in ​n​  , and​​
1538 THE AMERICAN ECONOMIC REVIEW JUNE 2018

v​N​​ (n)​ is negative and increasing in ​n​. Thus, the only possible BGP in this
region must be one with ​n(t ) = 0​. No interior BGP exists with ​n ∈ (0, ​ ~ n ρ ))​.
​(
_
Proposition 4 shows that for ​ρ < ​ρ  ​​ , a path for technology with ​n(t ) = 0​
yields balanced growth. Moreover, along this path all tasks are produced with
capital, which implies that ​​V​I​​ (t ) = ​VN​ ​​ (t ) = 0​. Thus, a path for ­technology
in which n​ (t ) = 0​is consistent with the equilibrium allocation of scientists.
The resulting BGP is an equilibrium with endogenous technology.
_ _ _
(ii) ρ ​ > ​ρ  ​​: Suppose ​n(t) ≤ ​  n ​(ρ)​. Then, we have ​n *(t ) = ​  n ​(ρ)​ and there-
_ _
fore ​​vN​ ​​ (n ) = ​vN​ ​(​n ​( ρ ))​ and ​​vI​​(n) = ​vI​​(​n ​(ρ))​ . Moreover, Lemma A3
_ _ _
implies that ​​κN​ ​​vN​ ​(​n ​( ρ )) > ​κI​​​vI​​(​n ​( ρ))​and ​​vI​​(​n ​( ρ )) = (g)​ with ​g​ small
~
(again because ​S < ​ S​​  ). Therefore, in this region this region we always have
that all scientists will be employed to create new tasks, and thus ​n​ ̇​ > 0​
_
(and is uniformly bounded away from zero). But this contradicts ​n(t) < ​n ​( ρ )​.
_
Suppose, instead, that ​n(t) > ​n ​( ρ )​. Then, Proposition 4 shows that the
­economy admits a BGP only if ​n(t) = n​. Thus, a necessary and sufficient
condition for an interior BGP is (29) in the text. Consequently, each i­nterior
_
BGP corresponds to a solution to this equation in (​​n ​( ρ ), 1)​. Lemma A3
_ _ _ _
shows that at ​​ n ​​  , ​​κN​ ​​vN​ ​(​n ​)​is above ​ ​κ​I​​vI​​(​n ​)​  , and ​κ ​I​​​vI​​(​n ​) = (g)​.
Moreover, when ​​κI​ ​​/​κN​ ​​  = 0​ , the entire curve ​​κN​ ​​ ​vN​ ​​ (n)​is above ​​κI​ ​​ ​vI​​​ (n)​.
As this ratio increases, the curve ​κ ​ ​ ​​ ​v​​​ (n)​rotates up, and eventually crosses
_I I
​​κN​ ​​ ​vN​ ​​ (n)​ at a point to the right of ​​  n ​(ρ)​. This defines the threshold ​​κ _​​. Above
_ _
this t­ hreshold, there exists another threshold ​​κ  ​​such that if ​κ ​ I​ ​​/​κN​ ​​  > ​κ  ​​  , there
is a unique intersection of ​​κI​ ​​ ​vI​​​ (n)​and ​​κN​ ​​ ​vN​ ​​ (n)​. (Note that one could have​​
_ _
_ κ ​ = ​κ​​.) By continuity, there exists ˆ ​​ ​such that, the thresholds ​​_
S​ κ ​​and ​​ ​​κ
ˆ ​κ​I​​ ​κN​ ​​
are defined for all ​S < ​ S​​ (recall that ​g = A ​ _ ​κI​ ​​ + ​κ​N​​  ​  )
S​ . It then follows that
~ˆ _
for ​S < min  ​{​, ​ ​ S S​}​​ and ​​κI​ ​​/​κN​ ​​  > ​κ  ​​  , there exists a unique BGP, which is
_
­interior and ­satisfies ​n(t ) = n*(t) = ​nB​ ​​ ∈ (​  n ​, 1)​. For ​S < min  { ​ ​ S​, ​ˆ
S​}​​ and​​
~
_ _
_κ ​ < ​κI​ ​​/​κN​ ​​  < ​κ  ​​ (provided that _ ​​κ ​ < ​κ  ​​), the economy admits multiple BGPs
~ˆ
with e­ ndogenous technology. Finally, for ​S < min  ​{​, ​ ​ S S​}​ and ​κ ​ ​I​​/​κN​ ​​  < ​_ κ ​​  ,
the only potential BGP is the corner one with ​n(t ) = 1​as in part (iv) of
Proposition 4. Because ​κ ​ N​ ​​ ​vN​ ​​ (1) > ​κI​ ​​ ​vI​​​ (1)​, this path for technology is
­consistent with the equilibrium allocation of scientists and provides a BGP
with endogenous technology.

Part 2: Stability Analysis: The stability analysis applies to the case in which​
_ ~ˆ _
ρ > ​ρ ​​  , ​S < min  ​{​, ​
​ S S​}​​, and ​κ
​ ​I​​/​κN​ ​​  > ​κ
 ​​. In this case, the economy admits a
_
unique BGP defined by ​n​ B​ ​​ ∈ (​  n ​( ρ ), 1)​. We denote by ​​cB​ ​​ , ​kB​ ​​​, and ​​LB​ ​​​the values of
(­normalized) consumption and capital, and employment in this BGP.

PROOF OF GLOBAL STABILITY WHEN ​θ = 0​:

Because ​θ = 0​ , we also have R
​ = ρ  + δ​ , and capital adjusts immediately and
its equilibrium stock only depends on ​n​, which becomes the unique state variable
of the model.
VOL. 108 NO. 6 ACEMOGLU AND RESTREPO: THE RACE BETWEEN MAN AND MACHINE 1539

v v˙ = 0

Stable arm

Figure A2. Phase Diagram and Global Saddle Path Stability when ​​θ = 0​​

Notes: The figure plots the locus for ​​​​​v˙ ​ = 0​​and the locus for n​​˙​ = 0​​. The unique BGP
is located at their interception.

Let v​ = ​κI​ ​​ ​vI​​​ − ​κN​ ​​ ​vN​ ​​​

. Now starting from any n​(0)​
, an equilibrium with
e­ ndogenous technology is given by the path of ​(n, v)​such that the evolution of the
state variable is given by

​n ​̇ = ​κN​ ​​ S − ( ​κN​ ​​ + ​κI​ ​​ )G​(v)​S;​

​

and the evolution of the difference of the normalized value functions, v​ ​, satisfies the
forward-looking differential equation

​ρv − ​v ​̇ = b​κI​ ​​​(​c​​  u​​​(ρ + δ)​​​  ζ−σ​− ​c​​  u​​​(​wI​​​)​​​  ζ−σ​)​− b​κN​ ​​​(​c​​  u​​​(​wN​ ​​)​​​  ζ−σ​− ​c​​  u​(​​ ρ + δ )​​​  ζ−σ​)​+ ( g )​

together with the transversality condition (27) holds.

When ​ g = 0​  , the locus for v​​˙​ = 0​crosses zero from below at a unique
point (recall that we are in the parameter region where there is a unique BGP).
By ­continuity there exists a threshold ​​S˘ ​​such that, for ​S < ​S˘ ​​, the locus for ​​v ​̇ = 0​
crosses zero from below at a unique point ​n​ B​ ​​​ , which denotes the BGP value for n​ (t)​
derived from (29).
We now analyze the stability properties of the system and show that the
BGP is globally saddle-path stable. Figure A2 presents the phase diagram of the sys-
tem in (​ v, n)​. The locus for ​v​  ​̇ = 0​crosses v​ = 0​at ​​nB​ ​​​from below only once. This
follows from the fact that ​​κI​ ​​ ​vI​​​ (n)​cuts ​​κN​ ​​ ​vN​ ​​ (n)​from below at ​​n​B​​​as shown in Figure 8.
The laws of motion of the two variables, ​v​and ​n​, take the form shown in the phase
diagram.34 This implies the existence of the unique stable arm, and also establishes

34
This can also be verified locally from the fact that the behavior of ​n​and ​ v​near the BGP can be ­approximated
by the linear system ​ ​ṅ ​ = − ( ​κN​ ​​ + ​κI​ ​​ )​G′ ​(0 ) Sv​and ​ ​v ​̇ = ρv − Q,​where ​Q > 0​denotes the derivative of
​− M ​κI​ ​​ ​c​​  u​​​(​w​I​​)​​​  ζ−σ​+ M ​κN​ ​​ ​c​​  u​​​( ​wN​ ​​ )​​​  ζ−σ​​with respect to n​ ​ (this derivative is positive because ​​κI​ ​​ ​vI​​​ (n)​cuts ​​κ​N​​ ​vN​ ​​ (n)​
from below at ​​n​B​​  ). Because the product of the eigenvalues of the characteristic polynomial of this system is
​− Q( ​κN​ ​+ ​κI​​)G′( 0) S < 0​ , there is one positive and one negative eigenvalue (and their sum is ρ ​ > 0​ , so the
­positive one is larger in absolute value).
1540 THE AMERICAN ECONOMIC REVIEW JUNE 2018

that there are no equilibrium paths that are not along this stable arm. In particular, all
paths above the stable arm feature v​​˙​ > 0​and eventually n​ → 0​and v​ → ∞​, and
since ​​v​N​​​is positive, ​​v​I​​ → ∞​. But this violates the ­transversality ­condition, (27).
Similarly, all paths below the stable arm feature v​​˙​ < 0​and e­ ventually ​n → 1​ and​
v → − ∞​ , and thus ​​v​N​​ → ∞​ , once again violating the transversality condition.

PROOF OF LOCAL STABILITY OF THE UNIQUE BGP WHEN ​θ > 0​  :

Online Appendix B shows that there exists a threshold ˇ ​​  such that the BGP in this
S​​
_
case is locally stable for ​
S < ​  ˇ​ , and thus the conclusions of the proposition follow
S ​
~ˆ ˘ ˇ
­setting ​​  S ​ = min  ​{​, ​
​ S S​, ​S​, ​ S ​}​​. ​∎​

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